From Coq Require Import ssreflect ssrfun ssrbool.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Require Import Setoid.
From Coq Require Import Ensembles.
From Coq.Logic Require Import FunctionalExtensionality.
From Coq.Logic Require Import PropExtensionality ClassicalFacts.

From stdpp Require Import base list sets propset.
New coercion path [Bool.Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]

From MatchingLogic Require Import
  Syntax
  Semantics
  DerivedOperators_Syntax
  DerivedOperators_Semantics
  monotonic
  Utils.Lattice
  Utils.stdpp_ext
  IndexManipulation
.

Import MatchingLogic.Syntax.Notations.
Import MatchingLogic.DerivedOperators_Syntax.Notations.


Section with_signature.
  Context {Σ : Signature}.

  Lemma eval_mu_lfp_fixpoint M ρ ϕ :
    well_formed_positive (patt_mu ϕ) ->
    let X := fresh_svar ϕ in
    let F := Fassoc ρ (svar_open 0 X ϕ) X in
    let Sfix := @eval Σ M ρ (patt_mu ϕ) in
    F Sfix = Sfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  let Sfix := eval ρ (mu , ϕ) in F Sfix = Sfix
  Proof.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  let Sfix := eval ρ (mu , ϕ) in F Sfix = Sfix
    simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
→ Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
    (fresh_svar ϕ) (eval ρ (mu , ϕ)) = eval ρ (mu , ϕ)
    remember (fresh_svar ϕ) as X.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
→ Fassoc ρ (svar_open 0 X ϕ) X (eval ρ (mu , ϕ)) =
  eval ρ (mu , ϕ)
    remember (Fassoc ρ (svar_open 0 X ϕ) X) as F.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
→ F (eval ρ (mu , ϕ)) = eval ρ (mu , ϕ)
    remember (@eval Σ M ρ (patt_mu ϕ)) as Sfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F Sfix = Sfix
    pose (OS := PropsetOrderedSet (Domain M)).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F Sfix = Sfix
    pose (L := PowersetLattice (Domain M)).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F Sfix = Sfix
    intros Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
F Sfix = Sfix

    assert (HFmono: MonotonicFunction F).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction F
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
F Sfix = Sfix
    {
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction F rewrite HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction (Fassoc ρ (svar_open 0 X ϕ) X) rewrite /Fassoc.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction
  (λ S : propset M,
     eval (update_svar_val X S ρ) (svar_open 0 X ϕ)) apply is_monotonic.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
well_formed_positive (mu , ϕ)
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in X ϕ apply Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in X ϕ
      rewrite HeqX.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in (fresh_svar ϕ) ϕ apply set_svar_fresh_is_fresh.
    }
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
F Sfix = Sfix

    assert (Ffix : Lattice.isFixpoint F (Lattice.LeastFixpointOf F)).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
isFixpoint F (LeastFixpointOf F)
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: isFixpoint F (LeastFixpointOf F)
F Sfix = Sfix
    {
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
isFixpoint F (LeastFixpointOf F) apply Lattice.LeastFixpoint_fixpoint.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
MonotonicFunction F apply HFmono.
    }
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: isFixpoint F (LeastFixpointOf F)
F Sfix = Sfix

    unfold isFixpoint in Ffix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
F Sfix = Sfix
    rewrite eval_mu_simpl in HeqSfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
(let X := fresh_svar ϕ in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' (svar_open 0 X ϕ)))
F Sfix = Sfix
    simpl in HeqSfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ))
F Sfix = Sfix
    unfold Fassoc in HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F =
(λ S : propset M,
   eval (update_svar_val X S ρ)
     (svar_open 0 X ϕ))
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ))
F Sfix = Sfix
    rewrite HeqX in HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ))
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
F Sfix = Sfix
    rewrite -HeqF in HeqSfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
HeqSfix: Sfix = LeastFixpointOf F
F Sfix = Sfix
    rewrite -HeqSfix in Ffix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
HeqSfix: Sfix = LeastFixpointOf F
Ffix: F Sfix = Sfix
F Sfix = Sfix
    apply Ffix.
  Qed.


  Lemma eval_mu_lfp_least M ρ ϕ S:
    well_formed_positive (patt_mu ϕ) ->
    let X := fresh_svar ϕ in
    let F := Fassoc ρ (svar_open 0 X ϕ) X in
    let Sfix := @eval Σ M ρ (patt_mu ϕ) in
    (F S) ⊆ S ->
    Sfix ⊆ S.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  let Sfix := eval ρ (mu , ϕ) in F S ⊆ S → Sfix ⊆ S
  Proof.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  let Sfix := eval ρ (mu , ϕ) in F S ⊆ S → Sfix ⊆ S
    simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
→ Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
    (fresh_svar ϕ) S ⊆ S → eval ρ (mu , ϕ) ⊆ S
    remember (fresh_svar ϕ) as X.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
→ Fassoc ρ (svar_open 0 X ϕ) X S ⊆ S
  → eval ρ (mu , ϕ) ⊆ S
    remember (Fassoc ρ (svar_open 0 X ϕ) X) as F.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F S ⊆ S → eval ρ (mu , ϕ) ⊆ S
    remember (@eval Σ M ρ (patt_mu ϕ)) as Sfix.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F S ⊆ S → Sfix ⊆ S
    pose (OS := PropsetOrderedSet (Domain M)).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F S ⊆ S → Sfix ⊆ S
    pose (L := PowersetLattice (Domain M)).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ → F S ⊆ S → Sfix ⊆ S
    intros Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
F S ⊆ S → Sfix ⊆ S

    assert (HFmono: MonotonicFunction F).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction F
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
F S ⊆ S → Sfix ⊆ S
    {
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction F rewrite HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction (Fassoc ρ (svar_open 0 X ϕ) X) rewrite /Fassoc.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
MonotonicFunction
  (λ S : propset M,
     eval (update_svar_val X S ρ) (svar_open 0 X ϕ)) apply is_monotonic.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
well_formed_positive (mu , ϕ)
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in X ϕ apply Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in X ϕ
      rewrite HeqX.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
svar_is_fresh_in (fresh_svar ϕ) ϕ apply set_svar_fresh_is_fresh.
    }
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
F S ⊆ S → Sfix ⊆ S

    assert (Hlfp: LeastFixpointOf F = Sfix).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
LeastFixpointOf F = Sfix
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
F S ⊆ S → Sfix ⊆ S
    {
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
LeastFixpointOf F = Sfix subst.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ))
LeastFixpointOf
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ)) = eval ρ (mu , ϕ) rewrite eval_mu_simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ))
LeastFixpointOf
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ)) =
(let X := fresh_svar ϕ in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' (svar_open 0 X ϕ))) simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ))
LeastFixpointOf
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ)) =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ)) unfold Fassoc.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction
  (Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
     (fresh_svar ϕ))
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ)) =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ)) reflexivity. }
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
F S ⊆ S → Sfix ⊆ S

    intros Hincl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S ⊆ S
Sfix ⊆ S

    pose proof (Hleast := LeastFixpoint_LesserThanPrefixpoint _ _ _ F S).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S ⊆ S
Hleast: leq (F S) S → leq (LeastFixpointOf F) S
Sfix ⊆ S
    simpl in Hleast.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S ⊆ S
Hleast: F S ⊆ S → LeastFixpointOf F ⊆ S
Sfix ⊆ S specialize (Hleast Hincl).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S ⊆ S
Hleast: LeastFixpointOf F ⊆ S
Sfix ⊆ S
    rewrite Hlfp in Hleast.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , ϕ)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S ⊆ S
Hleast: Sfix ⊆ S
Sfix ⊆ S apply Hleast.
  Qed.

  Lemma eval_mu_if_lfp M ρ ϕ Sfix :
    well_formed_positive (patt_mu ϕ) ->
    let X := fresh_svar ϕ in
    let F := Fassoc ρ (svar_open 0 X ϕ) X in
    (F Sfix) ⊆ Sfix ->
    (∀ S, (F S) ⊆ S -> Sfix ⊆ S) ->
    Sfix = @eval Σ M ρ (patt_mu ϕ).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  F Sfix ⊆ Sfix
  → (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
    → Sfix = eval ρ (mu , ϕ)
  Proof.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  F Sfix ⊆ Sfix
  → (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
    → Sfix = eval ρ (mu , ϕ)
    intros Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
let X := fresh_svar ϕ in
let F := Fassoc ρ (svar_open 0 X ϕ) X in
F Sfix ⊆ Sfix
→ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
  → Sfix = eval ρ (mu , ϕ) simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  Sfix ⊆ Sfix
→ (∀ S : propset M,
     Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
       (fresh_svar ϕ) S ⊆ S → Sfix ⊆ S)
  → Sfix = eval ρ (mu , ϕ)
    remember (fresh_svar ϕ) as X.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
Fassoc ρ (svar_open 0 X ϕ) X Sfix ⊆ Sfix
→ (∀ S : propset M,
     Fassoc ρ (svar_open 0 X ϕ) X S ⊆ S → Sfix ⊆ S)
  → Sfix = eval ρ (mu , ϕ)
    remember (Fassoc ρ (svar_open 0 X ϕ) X) as F.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
F Sfix ⊆ Sfix
→ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
  → Sfix = eval ρ (mu , ϕ)
    intros Hprefix Hleast.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
Sfix = eval ρ (mu , ϕ)
    rewrite eval_mu_simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
Sfix =
(let X := fresh_svar ϕ in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' (svar_open 0 X ϕ))) simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ))
    unfold Fassoc in HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F =
(λ S : propset M,
   eval (update_svar_val X S ρ)
     (svar_open 0 X ϕ))
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ)) rewrite HeqX in HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
Sfix =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ) S ρ)
       (svar_open 0 (fresh_svar ϕ) ϕ)) rewrite -HeqF.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
Sfix = LeastFixpointOf F
    apply LeastFixpoint_unique.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
leq (F Sfix) Sfix
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
∀ x : propset M, leq (F x) x → leq Sfix x {
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
leq (F Sfix) Sfix apply Hprefix. }
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
Hprefix: F Sfix ⊆ Sfix
Hleast: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
HeqF: F =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ) S ρ)
     (svar_open 0 (fresh_svar ϕ) ϕ))
∀ x : propset M, leq (F x) x → leq Sfix x apply Hleast.
  Qed.

  Lemma eval_mu_lfp_iff M ρ ϕ Sfix :
    well_formed_positive (patt_mu ϕ) ->
    let X := fresh_svar ϕ in
    let F := Fassoc ρ (svar_open 0 X ϕ) X in
    (
    (F Sfix) ⊆ Sfix /\
    (∀ S, (F S) ⊆ S -> Sfix ⊆ S)
    ) <-> Sfix = @eval Σ M ρ (patt_mu ϕ).
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  F Sfix ⊆ Sfix
  ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
  ↔ Sfix = eval ρ (mu , ϕ)
  Proof.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
well_formed_positive (mu , ϕ)
→ let X := fresh_svar ϕ in
  let F := Fassoc ρ (svar_open 0 X ϕ) X in
  F Sfix ⊆ Sfix
  ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
  ↔ Sfix = eval ρ (mu , ϕ)
    intros Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
let X := fresh_svar ϕ in
let F := Fassoc ρ (svar_open 0 X ϕ) X in
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
↔ Sfix = eval ρ (mu , ϕ) simpl.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  Sfix ⊆ Sfix
∧ (∀ S : propset M,
     Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
       (fresh_svar ϕ) S ⊆ S → Sfix ⊆ S)
↔ Sfix = eval ρ (mu , ϕ)
    remember (fresh_svar ϕ) as X.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
Fassoc ρ (svar_open 0 X ϕ) X Sfix ⊆ Sfix
∧ (∀ S : propset M,
     Fassoc ρ (svar_open 0 X ϕ) X S ⊆ S → Sfix ⊆ S)
↔ Sfix = eval ρ (mu , ϕ)
    remember (Fassoc ρ (svar_open 0 X ϕ) X) as F.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
↔ Sfix = eval ρ (mu , ϕ)
    remember (@eval Σ M ρ (patt_mu ϕ)) as Sfix'.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
↔ Sfix = Sfix'
    split.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
→ Sfix = Sfix'
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
Sfix = Sfix'
→ F Sfix ⊆ Sfix
  ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
    -
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S)
→ Sfix = Sfix' intros [H1 H2].
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H1: F Sfix ⊆ Sfix
H2: ∀ S : propset M, F S ⊆ S → Sfix ⊆ S
Sfix = Sfix' subst.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
H2: ∀ S : propset M,
  Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
    (fresh_svar ϕ) S ⊆ S → 
  Sfix ⊆ S
H1: Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
  (fresh_svar ϕ) Sfix ⊆ Sfix
Sfix = eval ρ (mu , ϕ)
      auto using eval_mu_if_lfp.
    -
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
Sfix = Sfix'
→ F Sfix ⊆ Sfix
  ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S) intros H.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
F Sfix ⊆ Sfix ∧ (∀ S : propset M, F S ⊆ S → Sfix ⊆ S) split.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
F Sfix ⊆ Sfix
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
∀ S : propset M, F S ⊆ S → Sfix ⊆ S
      +
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
F Sfix ⊆ Sfix subst.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  (eval ρ (mu , ϕ)) ⊆ eval ρ (mu , ϕ)
        match goal with
        | |- ?L ⊆ ?R => assert (H: L = R)
        end.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  (eval ρ (mu , ϕ)) = eval ρ (mu , ϕ)
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
H: Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) =
eval ρ (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) 
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) ⊆ 
eval ρ (mu , ϕ)
        apply eval_mu_lfp_fixpoint.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
well_formed_positive (mu , ϕ)
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
H: Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) =
eval ρ (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) 
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) ⊆ 
eval ρ (mu , ϕ) apply Hwfp.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
H: Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) =
eval ρ (mu , ϕ)
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  (eval ρ (mu , ϕ)) ⊆ eval ρ (mu , ϕ)
        rewrite H.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
H: Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ)
  (fresh_svar ϕ) (eval ρ (mu , ϕ)) =
eval ρ (mu , ϕ)
eval ρ (mu , ϕ) ⊆ eval ρ (mu , ϕ) apply reflexivity.
      +
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
∀ S : propset M, F S ⊆ S → Sfix ⊆ S intros S.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , ϕ)
X: svar
HeqX: X = fresh_svar ϕ
F: propset M → propset M
HeqF: F = Fassoc ρ (svar_open 0 X ϕ) X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , ϕ)
H: Sfix = Sfix'
S: propset M
F S ⊆ S → Sfix ⊆ S subst.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
S: propset M
Fassoc ρ (svar_open 0 (fresh_svar ϕ) ϕ) (fresh_svar ϕ)
  S ⊆ S → eval ρ (mu , ϕ) ⊆ S apply eval_mu_lfp_least.
Σ: Signature
M: Model
ρ: Valuation
ϕ: Pattern
Hwfp: well_formed_positive (mu , ϕ)
S: propset M
well_formed_positive (mu , ϕ) apply Hwfp.
  Qed.

  (* mu X. base \/ step X *)
  (* [Nats] = mu X. 0 \/ succ X *)
  (* [Nats] = \{ x | \ex x0,x1,..x_n . x0 \in 0 /\ x(i+1) \in succ xi }*)
  (*  0, 1, 2,... x*)
  Section inductive_generation.
    Context (base step : Pattern).

    Let patt_ind_gen_body := (patt_or (nest_mu base) (patt_app (nest_mu step) (patt_bound_svar 0))).
    Let patt_ind_gen_simple_body := (patt_or base (patt_app step (patt_free_svar (fresh_svar patt_ind_gen_body)))).
    
    Definition patt_ind_gen := patt_mu patt_ind_gen_body.

    Hypothesis (Hwfpbase : well_formed_positive base).
    Hypothesis (Hwfpstep : well_formed_positive step).

    Lemma patt_ind_gen_wfp:
      well_formed_positive patt_ind_gen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive patt_ind_gen
    Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive patt_ind_gen
      unfold patt_ind_gen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive (mu , patt_ind_gen_body) simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive (nest_mu base) && true,
    well_formed_positive (nest_mu step)
  & true]
      rewrite !andb_true_r.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive (nest_mu base)
  & well_formed_positive (nest_mu step)]
      rewrite !well_formed_positive_nest_mu_aux.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive base
  & well_formed_positive step]
      rewrite Hwfpbase.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    true
  & well_formed_positive step]
      rewrite Hwfpstep.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    true
  & true]
      rewrite !andb_true_r.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
no_negative_occurrence_db_b 0 patt_ind_gen_body

      cbn.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | ϕ₁ $ ϕ₂ =>
           no_negative_occurrence_db_b dbi ϕ₁ &&
           no_negative_occurrence_db_b dbi ϕ₂
       | ϕ₁ ---> ϕ₂ =>
           no_positive_occurrence_db_b dbi ϕ₁ &&
           no_negative_occurrence_db_b dbi ϕ₂
       | ex , ϕ' => no_negative_occurrence_db_b dbi ϕ'
       | mu , ϕ' =>
           no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
           if decide (n = dbi) then false else true
       | ϕ₁ $ ϕ₂ =>
           no_positive_occurrence_db_b dbi ϕ₁ &&
           no_positive_occurrence_db_b dbi ϕ₂
       | ϕ₁ ---> ϕ₂ =>
           no_negative_occurrence_db_b dbi ϕ₁ &&
           no_positive_occurrence_db_b dbi ϕ₂
       | ex , ϕ' => no_positive_occurrence_db_b dbi ϕ'
       | mu , ϕ' =>
           no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) 0 (nest_mu base) &&
    true, no_negative_occurrence_db_b 0 (nest_mu step)
  & true] fold no_negative_occurrence_db_b.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 (nest_mu base) &&
    true, no_negative_occurrence_db_b 0 (nest_mu step)
  & true]

      rewrite !no_negative_occurrence_db_nest_mu_aux.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& (if decide (0 < 0)
     then no_negative_occurrence_db_b 0 base
     else
      if decide (0 < 0 + 1)
      then true
      else no_negative_occurrence_db_b (0 - 1) base) &&
    true,
    if decide (0 < 0)
    then no_negative_occurrence_db_b 0 step
    else
     if decide (0 < 0 + 1)
     then true
     else no_negative_occurrence_db_b (0 - 1) step
  & true] simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
true
      auto.
    Qed.

    Lemma svar_open_patt_ind_gen_body_simpl M ρ X:
      svar_is_fresh_in X patt_ind_gen_body ->
      @eval Σ M ρ (svar_open 0 X patt_ind_gen_body)
      = @eval Σ M ρ (patt_or base (patt_app step (patt_free_svar X))).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
svar_is_fresh_in X patt_ind_gen_body
→ eval ρ (svar_open 0 X patt_ind_gen_body) =
  eval ρ (base or step $ patt_free_svar X)
    Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
svar_is_fresh_in X patt_ind_gen_body
→ eval ρ (svar_open 0 X patt_ind_gen_body) =
  eval ρ (base or step $ patt_free_svar X)
      intros Hfr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: svar_is_fresh_in X patt_ind_gen_body
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X)
      unfold svar_is_fresh_in in Hfr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X ∉ free_svars patt_ind_gen_body
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X) simpl in Hfr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X
∉ free_svars (nest_mu base) ∪ ∅
  ∪ (free_svars (nest_mu step) ∪ ∅)
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X)
      rewrite !simpl_free_svars in Hfr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X ∉ free_svars base ∪ free_svars step
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X)
      apply sets.not_elem_of_union in Hfr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: (X ∉ free_svars base) ∧ X ∉ free_svars step
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X)
      destruct Hfr as [Hfr1 Hfr2].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ (svar_open 0 X patt_ind_gen_body) =
eval ρ (base or step $ patt_free_svar X)

      rewrite /patt_ind_gen_body.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (svar_open 0 X
     (nest_mu base or nest_mu step $ BoundVarSugar.B0)) =
eval ρ (base or step $ patt_free_svar X)
      unfold svar_open.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (bsvar_subst
     (nest_mu base or nest_mu step $ BoundVarSugar.B0)
     (patt_free_svar X) 0) =
eval ρ (base or step $ patt_free_svar X)
      simpl_bsvar_subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (bsvar_subst (nest_mu base) (patt_free_svar X) 0
   or bsvar_subst (nest_mu step) (patt_free_svar X) 0 $
      bsvar_subst BoundVarSugar.B0 (patt_free_svar X)
        0) = eval ρ (base or step $ patt_free_svar X) simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (bsvar_subst (nest_mu base) (patt_free_svar X) 0
   or bsvar_subst (nest_mu step) (patt_free_svar X) 0 $
      patt_free_svar X) =
eval ρ (base or step $ patt_free_svar X)
      rewrite 2!eval_or_simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (bsvar_subst (nest_mu base) (patt_free_svar X) 0)
∪ eval ρ
    (bsvar_subst (nest_mu step) (patt_free_svar X) 0 $
     patt_free_svar X) =
eval ρ base ∪ eval ρ (step $ patt_free_svar X)
      unfold nest_mu.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ
  (bsvar_subst (nest_mu_aux 0 1 base)
     (patt_free_svar X) 0)
∪ eval ρ
    (bsvar_subst (nest_mu_aux 0 1 step)
       (patt_free_svar X) 0 $ patt_free_svar X) =
eval ρ base ∪ eval ρ (step $ patt_free_svar X) rewrite 2!nest_mu_same.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X ∉ free_svars base
Hfr2: X ∉ free_svars step
eval ρ base ∪ eval ρ (step $ patt_free_svar X) =
eval ρ base ∪ eval ρ (step $ patt_free_svar X)
      reflexivity.
    Qed.
    
    
    Section with_eval.
      Context (M : @Model Σ).
      Context (ρ : @Valuation Σ M).


      Let F := let X := fresh_svar patt_ind_gen_body in
               @Fassoc Σ M ρ (svar_open 0 X patt_ind_gen_body) X.
      (*
      Lemma svar_open_patt_ind_gen_body_assoc S:
        let X := fresh_svar patt_ind_gen_body in
        eval ρₑ (update_svar_val X S ρₛ) (svar_open 0 X patt_ind_gen_body)
        = F S.
      Proof. reflexivity.
             (*
        cbv zeta.
        rewrite svar_open_patt_ind_gen_body_simpl.
        { apply set_svar_fresh_is_fresh. }
        subst F. unfold Fassoc.
        rewrite svar_open_patt_ind_gen_body_simpl.
        { apply set_svar_fresh_is_fresh.  }
        reflexivity.*)
      Qed.
       *)

      (* I can imagine this lemma to be proven automatically. *)
      Lemma F_interp: F = λ A, (eval ρ base)
                                 ∪ (app_ext (eval ρ step) A).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
F =
(λ A : propset M,
   eval ρ base ∪ app_ext (eval ρ step) A)
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
F =
(λ A : propset M,
   eval ρ base ∪ app_ext (eval ρ step) A)
        unfold F.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Fassoc ρ
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     patt_ind_gen_body) (fresh_svar patt_ind_gen_body) =
(λ A : propset M,
   eval ρ base ∪ app_ext (eval ρ step) A) unfold Fassoc.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar patt_ind_gen_body) S
        ρ)
     (svar_open 0 (fresh_svar patt_ind_gen_body)
        patt_ind_gen_body)) =
(λ A : propset M,
   eval ρ base ∪ app_ext (eval ρ step) A) apply functional_extensionality.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
∀ x : propset M,
  eval
    (update_svar_val (fresh_svar patt_ind_gen_body) x
       ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       patt_ind_gen_body) =
  eval ρ base ∪ app_ext (eval ρ step) x
        intros A.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     patt_ind_gen_body) =
eval ρ base ∪ app_ext (eval ρ step) A rewrite svar_open_patt_ind_gen_body_simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  patt_ind_gen_body
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  (base
   or step $
      patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base ∪ app_ext (eval ρ step) A
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  patt_ind_gen_body apply set_svar_fresh_is_fresh. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  (base
   or step $
      patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base ∪ app_ext (eval ρ step) A

        rewrite eval_or_simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
∪ eval
    (update_svar_val (fresh_svar patt_ind_gen_body) A
       ρ)
    (step $
     patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base ∪ app_ext (eval ρ step) A
        rewrite eval_app_simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
∪ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step)
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ)
       (patt_free_svar (fresh_svar patt_ind_gen_body))) =
eval ρ base ∪ app_ext (eval ρ step) A
        rewrite eval_free_svar_simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
∪ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step)
    (svar_valuation
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) (fresh_svar patt_ind_gen_body)) =
eval ρ base ∪ app_ext (eval ρ step) A
        rewrite update_svar_val_same.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
∪ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base ∪ app_ext (eval ρ step) A
        rewrite eval_free_svar_independent.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval ρ base
∪ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base ∪ app_ext (eval ρ step) A
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) base
          eapply svar_is_fresh_in_richer.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
free_svars base ⊆ free_svars ?ϕ₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂
          2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ apply set_svar_fresh_is_fresh. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
free_svars base ⊆ free_svars patt_ind_gen_body
          solve_free_svars_inclusion 5.
        }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval ρ base
∪ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base ∪ app_ext (eval ρ step) A
        rewrite eval_free_svar_independent.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval ρ base ∪ app_ext (eval ρ step) A =
eval ρ base ∪ app_ext (eval ρ step) A
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
          eapply svar_is_fresh_in_richer.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
free_svars step ⊆ free_svars ?ϕ₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂
          2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ apply set_svar_fresh_is_fresh. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
free_svars step ⊆ free_svars patt_ind_gen_body
          solve_free_svars_inclusion 5.
        }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
A: propset M
eval ρ base ∪ app_ext (eval ρ step) A =
eval ρ base ∪ app_ext (eval ρ step) A
        reflexivity.
      Qed.

      Definition is_witnessing_sequence_old (m : Domain M) (l : list (Domain M)) :=
        (last l = Some m) /\
        (match l with
         | [] => False
         | m₀::ms => (m₀ ∈ @eval Σ M ρ base)
                     /\  (@Forall _
                                  (λ (x : (Domain M) * (Domain M)),
                                   let (old, new) := x in
                                   new ∈ 
                                  app_ext
                                    (@eval Σ M ρ step)
                                    {[ old ]}
                                 )
                                 (zip (m₀::ms) ms)
                         )
                           
         end).

      Definition is_witnessing_sequence (m : Domain M) (l : list (Domain M)) :=
        (∃ lst, last l = Some lst /\ lst ∈ @eval Σ M ρ base)
          /\
          hd_error l = Some m
          /\
          ((@Forall _ (uncurry (λ new old,
                              new ∈
                     app_ext
                       (@eval Σ M ρ step)
                       {[old]}
                    ))
                    (zip l (tail l))
          )).

      (* If we have a witnessing sequence x₁ x₂ ... xₙ xₙ₊₁ ... xlast
         and xₙ matches `base`, then xₙ xₙ₊₁ is a witnessing sequence, too.
       *)
      Lemma witnessing_sequence_middle (m : Domain M) (l : list (Domain M)) (n : nat) (m' : Domain M) :
        is_witnessing_sequence m l ->
        l !! n = Some m' ->
        m' ∈ @eval Σ M ρ base ->
        is_witnessing_sequence m' (drop n l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m': M
is_witnessing_sequence m l
→ l !! n = Some m'
  → m' ∈ eval ρ base
    → is_witnessing_sequence m' (drop n l)
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m': M
is_witnessing_sequence m l
→ l !! n = Some m'
  → m' ∈ eval ρ base
    → is_witnessing_sequence m' (drop n l)
        intros [[lst [Hlst Hbase] ] [Hhd Hfa] ] Hm' Hbase'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_witnessing_sequence m' (drop n l)
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
∃ lst : M,
  last (drop n l) = Some lst ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (drop n l) (tail (drop n l)))
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
∃ lst : M,
  last (drop n l) = Some lst ∧ lst ∈ eval ρ base exists lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last (drop n l) = Some lst ∧ lst ∈ eval ρ base split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last (drop n l) = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base
          rewrite -> last_drop with (x := lst).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some lst = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some lst = Some lst reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst apply Hlst. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l apply lookup_lt_is_Some_1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_Some (l !! n)
            rewrite Hm'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_Some (Some m') exists m'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some m' = Some m' reflexivity.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base
          apply Hbase.
        }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (drop n l) (tail (drop n l)))
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l)))
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m' apply hd_drop_lookup.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
l !! n = Some m' apply Hm'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l)))
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l))) rewrite tail_drop_comm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (drop n (tail l)))
          rewrite -zip_with_drop.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (drop n (zip l (tail l)))
          apply Forall_drop.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
          apply Hfa.
        }
      Qed.

      Lemma witnessing_sequence_tail (m : Domain M) (l : list (Domain M)) (m' : Domain M) :
        is_witnessing_sequence m l ->
        head (tail l) = Some m' ->
        is_witnessing_sequence m' (tail l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m': M
is_witnessing_sequence m l
→ head (tail l) = Some m'
  → is_witnessing_sequence m' (tail l)
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m': M
is_witnessing_sequence m l
→ head (tail l) = Some m'
  → is_witnessing_sequence m' (tail l)
        intros Hw Hhead.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m': M
Hw: is_witnessing_sequence m l
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)
        unfold is_witnessing_sequence in Hw.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m': M
Hw: (∃ lst : M, last l = Some lst ∧ lst ∈ eval ρ base)
∧ head l = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip l (tail l))
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)
        destruct Hw as [[lst [Hlst1 Hlst2] ] Hw].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)
        unfold is_witnessing_sequence.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
(∃ lst : M,
   last (tail l) = Some lst ∧ lst ∈ eval ρ base)
∧ head (tail l) = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (tail l) (tail (tail l)))
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
∃ lst : M,
  last (tail l) = Some lst ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (tail l) (tail (tail l)))
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
∃ lst : M,
  last (tail l) = Some lst ∧ lst ∈ eval ρ base exists lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst ∧ lst ∈ eval ρ base
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
lst ∈ eval ρ base 2: apply Hlst2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst
          eapply last_tail.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last l = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some ?m 2: apply Hhead.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last l = Some lst apply Hlst1.
        }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (tail l) (tail (tail l)))
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (tail l) (tail (tail l)))
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m' apply Hhead. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (tail l) (tail (tail l)))
        rewrite -tail_zip.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (tail (zip l (tail l)))
        apply Forall_tail.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
        destruct Hw as [Hw1 Hw2].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw1: head l = Some m
Hw2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
        apply Hw2.
      Qed.
      

      Lemma is_witnessing_sequence_iff_is_witnessing_sequence_old_reverse (m : Domain M) (l : list (Domain M)) :
        is_witnessing_sequence m l <-> is_witnessing_sequence_old m (reverse l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)
        assert (Feq: (uncurry
                        (flip
                           (λ new old : Domain M,
                                        new ∈
                                        app_ext
                                          (eval ρ step)
                                          {[old]})
                     ))
                     =  (λ x : Domain M * Domain M,
                               let (old, new) := x in
                               new ∈
                               app_ext
                                 (eval ρ step)
                                 {[old]})).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)
        
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]}) apply functional_extensionality.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
∀ x : M * M,
  uncurry
    (flip
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x =
  (let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]}) intros [x₁ x₂].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
x₁, x₂: M
uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) (x₁, x₂) =
(x₂ ∈ app_ext (eval ρ step) {[x₁]})
          reflexivity.
        }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)
        
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
→ is_witnessing_sequence_old m (reverse l)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence_old m (reverse l)
→ is_witnessing_sequence m l
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
→ is_witnessing_sequence_old m (reverse l) intros [[m' [Hlast Hbase] ] [Hhd Hfa] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last l = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
is_witnessing_sequence_old m (reverse l)
          destruct l as [|x l].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head [] = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse [])
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head (x :: l) = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l))
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head [] = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse []) simpl in Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: None = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse []) inversion Hhd. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head (x :: l) = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l))
          simpl in Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: Some x = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l)) inversion Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: Some x = Some m
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
H0: x = m
is_witnessing_sequence_old m (reverse (m :: l)) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Hhd: Some m = Some m
is_witnessing_sequence_old m (reverse (m :: l)) clear Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
is_witnessing_sequence_old m (reverse (m :: l))
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse (m :: l)) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
match reverse (m :: l) with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse (m :: l)) = Some m rewrite reverse_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse l ++ [m]) = Some m rewrite last_app_singleton.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Some m = Some m
            reflexivity.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
match reverse (m :: l) with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end

          (* reverse (m::l) <> nil *)
          destruct (reverse (m::l)) eqn:Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
False
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
False assert ( length (reverse (m::l)) = @length (Domain M) []).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length (reverse (m :: l)) = length []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (reverse (m :: l)) = length []
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length (reverse (m :: l)) = length [] rewrite Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length [] = length [] reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (reverse (m :: l)) = length []
False
            rewrite reverse_length in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (m :: l) = length []
False
            simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: S (length l) = 0
False inversion H.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)

          assert (Hm'd: m' = d).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
m' = d
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
Hm'd: m' = d
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
m' = d
            rewrite reverse_cons in Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
m' = d
            pose proof (H := @last_reverse_head _ l m).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
H: last (m :: l) = head (reverse l ++ [m])
m' = d
            rewrite Hlast in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
H: Some m' = head (reverse l ++ [m])
m' = d
            unfold reverse in Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = head (reverse l ++ [m])
m' = d
            rewrite Heq in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = head (d :: l0)
m' = d
            simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = Some d
m' = d
            inversion H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = Some d
H1: m' = d
d = d subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some d = Some d
d = d
            reflexivity.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
Hm'd: m' = d
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)
          subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base apply Hbase. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          clear Hbase.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          pose proof (Heq' := f_equal reverse Heq).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': reverse (reverse (m :: l)) = reverse (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          rewrite reverse_involutive in Heq'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          assert (Heq'' : l0 = tail (d::l0)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
l0 = tail (d :: l0)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
l0 = tail (d :: l0) reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)
          rewrite -> Heq'' at 2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) (tail (d :: l0))) clear Heq''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) (tail (d :: l0)))
          rewrite -Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))
          clear Heq' Heq l0.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))
          apply Forall_zip_flip_reverse in Hfa.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (flip
        (λ new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))
          clear Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hfa: Forall
  (uncurry
     (flip
        (λ new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))

          rewrite Feq in Hfa.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))
          apply Hfa.
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence_old m (reverse l)
→ is_witnessing_sequence m l intros H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
H: is_witnessing_sequence_old m (reverse l)
is_witnessing_sequence m l
          destruct H as [Hlast H2].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse l) = Some m
H2: match reverse l with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m l
          destruct l as [|x l].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse []) = Some m
H2: match reverse [] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse (x :: l)) = Some m
H2: match reverse (x :: l) with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m (x :: l)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse []) = Some m
H2: match reverse [] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m [] simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: None = Some m
H2: match reverse [] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m [] inversion Hlast. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse (x :: l)) = Some m
H2: match reverse (x :: l) with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence m (x :: l)

          destruct (reverse (x::l)) as [|y ys] eqn:Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Heq: reverse (x :: l) = []
Hlast: last [] = Some m
H2: False
is_witnessing_sequence m (x :: l)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
H2: y ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Heq: reverse (x :: l) = []
Hlast: last [] = Some m
H2: False
is_witnessing_sequence m (x :: l) inversion H2. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
H2: y ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)
          destruct H2 as [Hbase Hfa].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)

          rewrite -(reverse_involutive (y::ys)) in Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: reverse (x :: l) = reverse (reverse (y :: ys))
is_witnessing_sequence m (x :: l)
          apply (@inj _ _ (=) _ reverse) in Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
is_witnessing_sequence m (x :: l)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: reverse (x :: l) = reverse (reverse (y :: ys))
Inj eq eq reverse
          2: typeclasses eauto.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
is_witnessing_sequence m (x :: l)
          
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
∃ lst : M,
  last (x :: l) = Some lst ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (x :: l) (tail (x :: l)))
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
∃ lst : M,
  last (x :: l) = Some lst ∧ lst ∈ eval ρ base exists y.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (x :: l) = Some y ∧ y ∈ eval ρ base
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (x :: l) = Some y
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base
            rewrite Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (reverse (y :: ys)) = Some y
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base
            rewrite last_reverse.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (y :: ys) = Some y
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base
            reflexivity.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base
            apply Hbase.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (x :: l) (tail (x :: l)))

          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
            rewrite Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (reverse (y :: ys)) = Some m
            rewrite head_reverse.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (y :: ys) = Some m
            apply Hlast.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))

          assert (Hfeq': (λ new old : Domain M, new ∈ app_ext (eval ρ step) {[old]}) = flip (flip (λ new old : Domain M, new ∈ app_ext (eval ρ step) {[old]}))).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
(λ new old : M, new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
(λ new old : M, new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) apply functional_extensionality.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
∀ x : M,
  (λ old : M, x ∈ app_ext (eval ρ step) {[old]}) =
  flip
    (flip
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x intros x0.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0: M
(λ old : M, x0 ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) x0
            apply functional_extensionality.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0: M
∀ x : M,
  (x0 ∈ app_ext (eval ρ step) {[x]}) =
  flip
    (flip
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x0 x intros x1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0, x1: M
(x0 ∈ app_ext (eval ρ step) {[x1]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) x0 x1
            reflexivity.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
          rewrite Hfeq'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (flip
           (λ new old : M,
              new ∈ app_ext (eval ρ step) {[old]}))))
  (zip (x :: l) (tail (x :: l)))
          rewrite Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (flip
           (λ new old : M,
              new ∈ app_ext (eval ρ step) {[old]}))))
  (zip (reverse (y :: ys)) (tail (reverse (y :: ys))))
          rewrite -Forall_zip_flip_reverse.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (λ new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (y :: ys) (tail (y :: ys)))
          rewrite Feq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(λ x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (λ new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) (tail (y :: ys)))
          apply Hfa.
      Qed.

      Lemma witnessing_sequence_extend (m m' : Domain M) (l : list (Domain M)) :
        (is_witnessing_sequence m l
         /\ (∃ step', step' ∈ eval ρ step /\ m' ∈ app_interp step' m)
        ) <-> (is_witnessing_sequence m' (m'::l) /\ l ≠ []).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
↔ is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
↔ is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
→ is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
→ is_witnessing_sequence m l
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
→ is_witnessing_sequence m' (m' :: l) ∧ l ≠ [] intros [[[lst [Hlst Hbase] ] [Hhd Hwit] ] [step' [Hstep' Hm'] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
is_witnessing_sequence m' (m' :: l)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
l ≠ []
          2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
l ≠ [] destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head [] = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
[] ≠ []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
d :: l ≠ [] simpl in Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hhd: None = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
[] ≠ []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
d :: l ≠ [] inversion Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
d :: l ≠ [] discriminate. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
is_witnessing_sequence m' (m' :: l)
          move: m Hhd m' step' Hm' Hstep'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
∀ m : M,
  head l = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m' (m' :: l)
          induction l; intros m Hhd m' step' Hm' Hstep'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: head [] = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m']
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: head (a :: l) = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l)
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: head [] = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m'] simpl in Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: None = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m'] inversion Hhd.
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: head (a :: l) = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l) simpl in Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: Some a = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l) inversion Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: Some a = Some m
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H0: a = m
is_witnessing_sequence m' (m' :: m :: l) subst a.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
l: list M
lst, m: M
Hhd: Some m = Some m
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: m :: l) clear Hhd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
l: list M
lst, m: M
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: m :: l)
            unfold is_witnessing_sequence.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
l: list M
lst, m: M
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  → ∀ m : M,
      head l = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last (m' :: m :: l) = Some lst ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: l) = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: l) (tail (m' :: m :: l)))
            destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst, m: M
Hlst: last [m] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m]))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last (m' :: m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: d :: l) = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: d :: l)
         (tail (m' :: m :: d :: l)))
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst, m: M
Hlst: last [m] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m])) simpl in Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst, m: M
Hlst: Some m = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m])) inversion Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst, m: M
Hlst: Some m = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H0: m = lst
(∃ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst])) subst m.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hlst: Some lst = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
(∃ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst])) clear Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
(∃ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst]))
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
∃ lst0 : M,
  last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
head [m'; lst] = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [m'; lst] (tail [m'; lst]))
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
∃ lst0 : M,
  last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base exists lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
last [m'; lst] = Some lst ∧ lst ∈ eval ρ base simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Some lst = Some lst ∧ lst ∈ eval ρ base split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Some lst = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base reflexivity.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base apply Hbase. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
head [m'; lst] = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [m'; lst] (tail [m'; lst]))
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Some m' = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    [(m', lst)] split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Some m' = Some m'
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  [(m', lst)] reflexivity.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  [(m', lst)] apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', lst)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) []
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', lst)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[lst]} ∧ m' ∈ app_interp step' re
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] exists lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
step' ∈ eval ρ step
∧ lst ∈ {[lst]} ∧ m' ∈ app_interp step' lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]} ∧ m' ∈ app_interp step' lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
              apply Hstep'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]} ∧ m' ∈ app_interp step' lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
m' ∈ app_interp step' lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] constructor.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
m' ∈ app_interp step' lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) [] apply Hm'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
              apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  → ∀ m : M,
      head [] = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp step' lst
Hstep': step' ∈ eval ρ step
True exact I.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
(∃ lst : M,
   last (m' :: m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: d :: l) = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: d :: l)
         (tail (m' :: m :: d :: l)))
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
∃ lst : M,
  last (m' :: m :: d :: l) = Some lst
  ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (m' :: m :: d :: l)
       (tail (m' :: m :: d :: l)))
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
∃ lst : M,
  last (m' :: m :: d :: l) = Some lst
  ∧ lst ∈ eval ρ base exists lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
last (m' :: m :: d :: l) = Some lst
∧ lst ∈ eval ρ base split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
last (m' :: m :: d :: l) = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
last (d :: l) = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base simpl in Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
last (d :: l) = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base apply Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base apply Hbase. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (m' :: m :: d :: l)
       (tail (m' :: m :: d :: l)))
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l)))
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m' reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l)))
            simpl in Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l))) simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m', m)
   :: (m, d)
      :: match l with
         | [] => []
         | x2 :: l2 => (d, x2) :: zip l l2
         end)
            apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  → ∀ m : M,
      head (d :: l) = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
            simpl in IHl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end
  → ∀ m : M,
      Some d = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end) simpl in Hlst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
→ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end
  → ∀ m : M,
      Some d = Some m
      → ∀ m' step' : M,
          m' ∈ app_interp step' m
          → step' ∈ eval ρ step
            → is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
            specialize (IHl Hlst).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
→ ∀ m : M,
    Some d = Some m
    → ∀ m' step' : M,
        m' ∈ app_interp step' m
        → step' ∈ eval ρ step
          → is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
            inversion Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
→ ∀ m : M,
    Some d = Some m
    → ∀ m' step' : M,
        m' ∈ app_interp step' m
        → step' ∈ eval ρ step
          → is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x = (m, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
→ ∀ m : M,
    Some d = Some m
    → ∀ m' step' : M,
        m' ∈ app_interp step' m
        → step' ∈ eval ρ step
          → is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end) clear Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
IHl: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
→ ∀ m : M,
    Some d = Some m
    → ∀ m' step' : M,
        m' ∈ app_interp step' m
        → step' ∈ eval ρ step
          → is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
            specialize (IHl H2).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end) clear Hlst H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m) exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp step' re exists m.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
step' ∈ eval ρ step
∧ m ∈ {[m]} ∧ m' ∈ app_interp step' m split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
m ∈ {[m]} ∧ m' ∈ app_interp step' m apply Hstep'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
m ∈ {[m]} ∧ m' ∈ app_interp step' m split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
m ∈ {[m]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
m' ∈ app_interp step' m constructor.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
m' ∈ app_interp step' m apply Hm'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
            apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d)
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d) apply H1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m : M,
  Some d = Some m
  → ∀ m' step' : M,
      m' ∈ app_interp step' m
      → step' ∈ eval ρ step
        → is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            specialize (IHl d erefl).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m' step' : M,
  m' ∈ app_interp step' d
  → step' ∈ eval ρ step
    → is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            destruct H1 as [step'' [d'' [Hstep'' [Hd'' Hstep''d''] ] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m' step' : M,
  m' ∈ app_interp step' d
  → step' ∈ eval ρ step
    → is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'', d'': M
Hstep'': step'' ∈ eval ρ step
Hd'': d'' ∈ {[d]}
Hstep''d'': m ∈ app_interp step'' d''
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            inversion Hd''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m' step' : M,
  m' ∈ app_interp step' d
  → step' ∈ eval ρ step
    → is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'', d'': M
Hstep'': step'' ∈ eval ρ step
Hd'': d'' ∈ {[d]}
Hstep''d'': m ∈ app_interp step'' d''
H: d = d''
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d'', x2) :: zip l l2
  end subst d''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m' step' : M,
  m' ∈ app_interp step' d
  → step' ∈ eval ρ step
    → is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Hd'': d ∈ {[d]}
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end clear Hd''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: ∀ m' step' : M,
  m' ∈ app_interp step' d
  → step' ∈ eval ρ step
    → is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            specialize (IHl m step'' Hstep''d'' Hstep'').
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: is_witnessing_sequence m (m :: d :: l)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            unfold is_witnessing_sequence in IHl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: (∃ lst : M,
   last (m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m :: d :: l) = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m :: d :: l) (tail (m :: d :: l)))
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            simpl in IHl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: (∃ lst : M,
   last (d :: l) = Some lst ∧ lst ∈ eval ρ base)
∧ Some m = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      ((m, d)
       :: match l with
          | [] => []
          | x2 :: l2 => (d, x2) :: zip l l2
          end)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            destruct IHl as [_ [_ Hforall] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
            inversion Hforall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x = (m, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp step'' d
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, d)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end apply H2.
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
l: list M
is_witnessing_sequence m' (m' :: l) ∧ l ≠ []
→ is_witnessing_sequence m l
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)            
      Abort.
            
          


      Lemma witnessing_sequence_old_extend
            (m x m' : Domain M) (l : list (Domain M)):
        (is_witnessing_sequence_old m (x::l) /\
        ∃ step', (step' ∈ eval ρ step /\
        m' ∈ app_interp step' m)) <->
        (last (x::l) = Some m /\ is_witnessing_sequence_old m' ((x::l) ++ [m'])).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
↔ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
↔ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
→ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
→ is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
→ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
          intros [Hwit [step' [Hstep' Hm'] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit: is_witnessing_sequence_old m (x :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
          destruct Hwit as [Hwit1 [Hwit2 Hwit3] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last (x :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
is_witnessing_sequence_old m' ((x :: l) ++ [m'])
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last (x :: l) = Some m apply Hwit1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
is_witnessing_sequence_old m' ((x :: l) ++ [m'])
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last ((x :: l) ++ [m']) = Some m'
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
last ((x :: l) ++ [m']) = Some m' apply last_snoc. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
          simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
    end
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ eval ρ base apply Hwit2. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
          
          destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(x, m')] apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) [] split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[x]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] 2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
True exact I. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[x]}
            exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[x]} ∧ m' ∈ app_interp step' re exists x.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step
∧ x ∈ {[x]} ∧ m' ∈ app_interp step' x split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ {[x]} ∧ m' ∈ app_interp step' x
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step apply Hstep'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ {[x]} ∧ m' ∈ app_interp step' x
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ {[x]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' x
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x ∈ {[x]} constructor. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' x
            simpl in Hwit1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: Some x = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' x inversion Hwit1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit1: Some x = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
H0: x = m
m' ∈ app_interp step' m subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hwit2: m ∈ eval ρ base
Hwit1: Some m = Some m
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' m apply Hm'.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
          simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
      end)
          apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
          simpl in Hwit3.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end inversion Hwit3.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
x0: (M * M)%type
l0: list (M * M)
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x0 = (x, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end clear Hwit3.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
          rename H1 into Hd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end rename H2 into Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]} apply Hd. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end clear Hd.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end

          move: d Hwit1 Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
        end
          induction l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
∀ d : M,
  last [x; d] = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match [] ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip ([] ++ [m']) l2
        end
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
∀ d : M,
  last (x :: d :: a :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      ((d, a) :: zip (a :: l) l)
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match (a :: l) ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip ((a :: l) ++ [m']) l2
        end
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
∀ d : M,
  last [x; d] = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match [] ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip ([] ++ [m']) l2
        end intros D Hm _.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
D: M
Hm: last [x; D] = Some m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (D, x2) :: zip ([] ++ [m']) l2
  end
            simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
D: M
Hm: last [x; D] = Some m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(D, m')] simpl in Hm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
D: M
Hm: Some D = Some m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(D, m')] inversion Hm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
D: M
Hm: Some D = Some m
H0: D = m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')] subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Hm: Some m = Some m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')]
            clear Hm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')]
            apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[m]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[m]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
True exact I. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_ext (eval ρ step) {[m]}
            exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp step' re exists m.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step
∧ m ∈ {[m]} ∧ m' ∈ app_interp step' m split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m ∈ {[m]} ∧ m' ∈ app_interp step' m
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
step' ∈ eval ρ step apply Hstep'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m ∈ {[m]} ∧ m' ∈ app_interp step' m
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m ∈ {[m]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' m
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m ∈ {[m]} constructor. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
m' ∈ app_interp step' m
            apply Hm'.
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
∀ d : M,
  last (x :: d :: a :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      ((d, a) :: zip (a :: l) l)
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match (a :: l) ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip ((a :: l) ++ [m']) l2
        end intros d Hlast Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (a :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip ((a :: l) ++ [m']) l2
  end simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end)
            inversion Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
x0: (M * M)%type
l0: list (M * M)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
H: x0 = (d, a)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (a, x2) :: zip l l2
end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end)
            apply Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
    end
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
  end
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]} apply H1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
  end
            apply IHl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (x :: a :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (x :: a :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (a :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end apply Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp step' m
IHl: ∀ d : M,
  last (x :: d :: l) = Some m
  → Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    → Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
            apply H2.
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
→ is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) intros [Hlast H].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: is_witnessing_sequence_old m' ((x :: l) ++ [m'])
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          unfold is_witnessing_sequence in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: is_witnessing_sequence_old m' ((x :: l) ++ [m'])
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          destruct H as [_ H].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          remember (length l) as len.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Heqlen: len = length l
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          assert (Hlen: length l <= len).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Heqlen: len = length l
length l ≤ len
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Heqlen: len = length l
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Heqlen: len = length l
length l ≤ len lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Heqlen: len = length l
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          clear Heqlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
len: nat
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: x ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
    end
len: nat
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          destruct H as [Hbase Hall].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          unfold is_witnessing_sequence.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
is_witnessing_sequence_old m (x :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          apply and_assoc.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
last (x :: l) = Some m
∧ (x ∈ eval ρ base
   ∧ Forall
       (λ x : M * M,
          let (old, new) := x in
          new ∈ app_ext (eval ρ step) {[old]})
       (zip (x :: l) l))
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
last (x :: l) = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
(x ∈ eval ρ base
 ∧ Forall
     (λ x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     (zip (x :: l) l))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
last (x :: l) = Some m apply Hlast. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
(x ∈ eval ρ base
 ∧ Forall
     (λ x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     (zip (x :: l) l))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          apply and_assoc.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
x ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (x :: l) l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
x ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
x ∈ eval ρ base apply Hbase. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          clear Hbase.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≤ len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
          
          move: x l Hlast Hall Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
          induction len.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ 0
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ S len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ 0
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m) intros x l Hlast Hall Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
Hlen: length l ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: length (d :: l) ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: length (d :: l) ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: S (length l) ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) lia.  }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
True exact I. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
            inversion Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
x0: (M * M)%type
l: list (M * M)
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x0 = (x, m')
H0: l = []
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≤ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m clear Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≤ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m clear H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≤ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
            destruct H1 as [step' [m'' [Hstep' [Hm'm'' Hstep'm''] ] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≤ 0
step', m'': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[x]}
Hstep'm'': m' ∈ app_interp step' m''
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
            inversion Hm'm''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≤ 0
step', m'': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[x]}
Hstep'm'': m' ∈ app_interp step' m''
H: x = m''
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[m'']}
Hstep'm'': m' ∈ app_interp step' m''
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m clear Hm'm''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
            exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
step' ∈ eval ρ step ∧ m' ∈ app_interp step' m split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m apply Hstep'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m
            simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: Some m'' = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m inversion Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m', m'': M
Hlast: Some m'' = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m''
H0: m'' = m
m' ∈ app_interp step' m subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
Hlast: Some m = Some m
Hlen: length [] ≤ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp step' m
m' ∈ app_interp step' m
            apply Hstep'm''.
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ S len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m) intros x l Hlast Hall Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
Hlen: length l ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            destruct l as [|x' l'].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (x' :: l') ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((x' :: l') ++ [m']) l2
  end
Hlen: length (x' :: l') ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
Hlast: last [x] = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
Hlast: Some x = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) inversion Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
Hlast: Some x = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
H0: x = m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) clear Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x: M
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
H0: x = m
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (m, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
              simpl in Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
              inversion Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≤ S len
x: (M * M)%type
l: list (M * M)
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x = (m, m')
H0: l = []
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) clear Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
              clear H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
True exact I. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
              destruct H1 as [step' Hstep'].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step': M
Hstep': ∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp step' re
∃ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
              exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step': M
Hstep': ∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp step' re
step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
              destruct Hstep' as [m'' [Hstep' [Hmm'' Hm''] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
step' ∈ eval ρ step ∧ m' ∈ app_interp step' m
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
step' ∈ eval ρ step apply Hstep'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m
              inversion Hmm''.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
Hlen: length [] ≤ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp step' m''
H: m = m''
m' ∈ app_interp step' m'' subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m': M
len: nat
m'': M
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m''
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m'')
Hlen: length [] ≤ S len
step': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m'']}
Hm'': m' ∈ app_interp step' m''
m' ∈ app_interp step' m''
              apply Hm''.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (x' :: l') ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((x' :: l') ++ [m']) l2
  end
Hlen: length (x' :: l') ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            simpl in Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≤ S len
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            inversion Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≤ S len
x0: (M * M)%type
l: list (M * M)
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
H: x0 = (x, x')
H0: l =
match l' ++ [m'] with
| [] => []
| x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m) clear Hall.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
IHlen: ∀ (x : M) (l : list M),
  last (x :: l) = Some m
  → Forall
      (λ x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    → length l ≤ len
      → Forall
          (λ x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (∃ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            specialize (IHlen x' l' Hlast H2).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            rewrite Forall_cons.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
(x' ∈ app_ext (eval ρ step) {[x]}
 ∧ Forall
     (λ x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     match l' with
     | [] => []
     | x2 :: l2 => (x', x2) :: zip l' l2
     end)
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            apply and_assoc.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]} apply H1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp step' m)
            apply IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
length l' ≤ len
            simpl in Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≤ len
→ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: S (length l') ≤ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
length l' ≤ len lia.
      Qed.

      Definition witnessed_elements_old : propset (Domain M) :=
        PropSet (λ m, ∃ l, is_witnessing_sequence_old m l).

      Lemma witnessed_elements_old_prefixpoint : (F witnessed_elements_old) ⊆ witnessed_elements_old.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
F witnessed_elements_old ⊆ witnessed_elements_old
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
F witnessed_elements_old ⊆ witnessed_elements_old
        rewrite elem_of_subseteq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
∀ x : M,
  x ∈ F witnessed_elements_old
  → x ∈ witnessed_elements_old intros x Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x ∈ F witnessed_elements_old
x ∈ witnessed_elements_old
        unfold F in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ Fassoc ρ
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       patt_ind_gen_body)
    (fresh_svar patt_ind_gen_body)
    witnessed_elements_old
x ∈ witnessed_elements_old unfold Fassoc in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       patt_ind_gen_body)
x ∈ witnessed_elements_old unfold svar_open in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (bsvar_subst patt_ind_gen_body
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)) 0)
x ∈ witnessed_elements_old simpl in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    ((bsvar_subst (nest_mu base)
        (patt_free_svar
           (fresh_svar patt_ind_gen_body)) 0 --->
      ⊥) --->
     bsvar_subst (nest_mu step)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)) 0 $
     patt_free_svar
       (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
        rewrite eval_or_simpl in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (bsvar_subst (nest_mu base)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)) 0)
  ∪ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      (bsvar_subst (nest_mu step)
         (patt_free_svar
            (fresh_svar patt_ind_gen_body)) 0 $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
        fold (svar_open 0 (fresh_svar patt_ind_gen_body) (nest_mu base)) in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
  ∪ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      (bsvar_subst (nest_mu step)
         (patt_free_svar
            (fresh_svar patt_ind_gen_body)) 0 $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
        fold (svar_open 0 (fresh_svar patt_ind_gen_body) (nest_mu step)) in Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
  ∪ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      (svar_open 0 (fresh_svar patt_ind_gen_body)
         (nest_mu step) $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
        destruct Hx.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ witnessed_elements_old
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step) $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ witnessed_elements_old unfold Ensembles.In in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ witnessed_elements_old
          unfold witnessed_elements_old.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x
∈ {[ m | ∃ l : list M, is_witnessing_sequence_old m l ]}
          exists [x].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
is_witnessing_sequence_old x [x] unfold is_witnessing_sequence_old.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
last [x] = Some x
∧ x ∈ eval ρ base
  ∧ Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      (zip [x] [])
          simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
Some x = Some x
∧ x ∈ eval ρ base
  ∧ Forall
      (λ x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
Some x = Some x
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
Some x = Some x reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []
          split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
          2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [] constructor. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
          rewrite eval_free_svar_independent in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     (nest_mu base))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval ρ
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     (nest_mu base))
            eapply svar_is_fresh_in_richer.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
free_svars
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     (nest_mu base)) ⊆ free_svars ?ϕ₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ 2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ auto. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
free_svars
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     (nest_mu base)) ⊆ free_svars patt_ind_gen_body
            unfold svar_open.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
free_svars
  (bsvar_subst (nest_mu base)
     (patt_free_svar (fresh_svar patt_ind_gen_body)) 0)
⊆ free_svars patt_ind_gen_body
            solve_free_svars_inclusion 5.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval ρ
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base
          simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval ρ
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu base))
x ∈ eval ρ base unfold svar_open in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval ρ
    (bsvar_subst (nest_mu base)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)) 0)
x ∈ eval ρ base
          rewrite nest_mu_same in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x ∈ eval ρ base
x ∈ eval ρ base auto.
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step) $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old unfold Ensembles.In in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step) $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
          rewrite eval_app_simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (svar_open 0 (fresh_svar patt_ind_gen_body)
          (nest_mu step)))
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)))
x ∈ witnessed_elements_old
          rewrite eval_free_svar_simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (svar_open 0 (fresh_svar patt_ind_gen_body)
          (nest_mu step)))
    (svar_valuation
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
          rewrite update_svar_val_same in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (svar_open 0 (fresh_svar patt_ind_gen_body)
          (nest_mu step))) witnessed_elements_old
x ∈ witnessed_elements_old
          unfold app_ext in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ {[ e | ∃ le re : M,
           le
           ∈ eval
               (update_svar_val
                  (fresh_svar patt_ind_gen_body)
                  witnessed_elements_old ρ)
               (svar_open 0
                  (fresh_svar patt_ind_gen_body)
                  (nest_mu step))
           ∧ re ∈ witnessed_elements_old
             ∧ e ∈ app_interp le re ]}
x ∈ witnessed_elements_old
          destruct H as [step' [m [H1 [H2 Happ] ] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
H2: m ∈ witnessed_elements_old
Happ: x ∈ app_interp step' m
x ∈ witnessed_elements_old
          unfold witnessed_elements_old in H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
H2: m
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l ]}
Happ: x ∈ app_interp step' m
x ∈ witnessed_elements_old
          destruct H2 as [l Hl].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp step' m
x ∈ witnessed_elements_old

          unfold witnessed_elements_old.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp step' m
x
∈ {[ m | ∃ l : list M, is_witnessing_sequence_old m l ]}
          exists (l ++ [x]).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x (l ++ [x])

          (* `l` is not empty *)
          destruct l as [|m₀ l'] eqn:Heql.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ([] ++ [x])
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ([] ++ [x]) unfold is_witnessing_sequence in Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ([] ++ [x]) destruct Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Heql: l = []
H: last [] = Some m
H0: False
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ([] ++ [x]) simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
Heql: l = []
H: None = Some m
H0: False
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ([] ++ [x]) inversion H. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (svar_open 0 (fresh_svar patt_ind_gen_body)
       (nest_mu step))
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])

          unfold svar_open in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (bsvar_subst (nest_mu step)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)) 0)
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
is_witnessing_sequence_old x ((m₀ :: l') ++ [x]) rewrite nest_mu_same in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ) step
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          rewrite eval_free_svar_independent in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step' ∈ eval ρ step
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
            eapply svar_is_fresh_in_richer.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
free_svars step ⊆ free_svars ?ϕ₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?ϕ₂ apply set_svar_fresh_is_fresh. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
free_svars step ⊆ free_svars patt_ind_gen_body
            solve_free_svars_inclusion 2.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step' ∈ eval ρ step
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])

          epose proof (P := @witnessing_sequence_old_extend _ _ _ _).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step' ∈ eval ρ step
P: is_witnessing_sequence_old ?m (?x :: ?l)
∧ (∃ step'0 : M,
     step'0 ∈ eval ρ step
     ∧ ?m' ∈ app_interp step'0 ?m)
↔ last (?x :: ?l) = Some ?m
  ∧ is_witnessing_sequence_old 
      ?m' ((?x :: ?l) ++ [?m'])
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          destruct P as [P _].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step' ∈ eval ρ step
P: is_witnessing_sequence_old ?m (?x :: ?l)
∧ (∃ step'0 : M,
     step'0 ∈ eval ρ step
     ∧ ?m' ∈ app_interp step'0 ?m)
→ last (?x :: ?l) = Some ?m
  ∧ is_witnessing_sequence_old 
      ?m' ((?x :: ?l) ++ [?m'])
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          specialize (P (conj Hl (@ex_intro _ _ step' (conj H1 Happ)))).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x, step', m: M
l: list M
m₀: M
l': list M
Heql: l = m₀ :: l'
Hl: is_witnessing_sequence_old m (m₀ :: l')
Happ: x ∈ app_interp step' m
H1: step' ∈ eval ρ step
P: last (m₀ :: l') = Some m
∧ is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
is_witnessing_sequence_old x ((m₀ :: l') ++ [x])
          apply P.
      Qed.

      Lemma interp_included_in_witnessed_elements_old:
        (@eval Σ M ρ patt_ind_gen) ⊆ witnessed_elements_old.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen ⊆ witnessed_elements_old
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen ⊆ witnessed_elements_old
        apply eval_mu_lfp_least.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
well_formed_positive (mu , patt_ind_gen_body)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Fassoc ρ
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     patt_ind_gen_body) (fresh_svar patt_ind_gen_body)
  witnessed_elements_old ⊆ witnessed_elements_old
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
well_formed_positive (mu , patt_ind_gen_body) apply patt_ind_gen_wfp. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Fassoc ρ
  (svar_open 0 (fresh_svar patt_ind_gen_body)
     patt_ind_gen_body) (fresh_svar patt_ind_gen_body)
  witnessed_elements_old ⊆ witnessed_elements_old
        apply witnessed_elements_old_prefixpoint.
      Qed.

      Lemma eval_patt_ind_gen_fix :
        let Sfix := eval ρ patt_ind_gen in
        F Sfix = Sfix.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
let Sfix := eval ρ patt_ind_gen in F Sfix = Sfix
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
let Sfix := eval ρ patt_ind_gen in F Sfix = Sfix
        apply eval_mu_lfp_fixpoint.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
well_formed_positive (mu , patt_ind_gen_body)
        apply patt_ind_gen_wfp.
      Qed.
      
      
      Definition witnessed_elements_old_of_max_len len : propset (Domain M) :=
        PropSet (λ m, ∃ l, is_witnessing_sequence_old m l /\ length l <= len).

      Lemma witnessed_elements_old_of_max_len_included_in_interp len:
        (witnessed_elements_old_of_max_len (S len)) ⊆ (@eval Σ M ρ patt_ind_gen).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
        induction len.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old_of_max_len 1
⊆ eval ρ patt_ind_gen
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
⊆ eval ρ patt_ind_gen
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old_of_max_len 1
⊆ eval ρ patt_ind_gen unfold Included.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old_of_max_len 1
⊆ eval ρ patt_ind_gen intros m.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
→ m ∈ eval ρ patt_ind_gen
          rewrite -eval_patt_ind_gen_fix.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
→ m ∈ F (eval ρ patt_ind_gen)
          unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
→ m ∈ F (eval ρ patt_ind_gen)
          intros H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m ∈ F (eval ρ patt_ind_gen)
          unfold witnessed_elements_old_of_max_len.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m ∈ F (eval ρ patt_ind_gen)
          rewrite F_interp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct H as [l [Hwit Hlen] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          unfold is_witnessing_sequence in Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct Hwit as [Hlast Hm].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
Hlast: last l = Some m
Hm: match l with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
Hlen: length l ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlast: last [] = Some m
Hm: False
Hlen: length [] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlast: last [] = Some m
Hm: False
Hlen: length [] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlast: None = Some m
Hm: False
Hlen: length [] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) inversion Hlast. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
Hlast: last [d] = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: length (d :: d0 :: l) ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: length (d :: d0 :: l) ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) simpl in Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (S (length l)) ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
Hlast: last [d] = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
Hlast: Some d = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≤ 1
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) inversion Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
Hlast: Some d = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≤ 1
H0: d = m
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m, d: M
Hm: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≤ 1
H0: d = m
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlen: length [m] ≤ 1
Hm: m ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [m] [])
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct Hm as [Hm _].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlen: length [m] ≤ 1
Hm: m ∈ eval ρ base
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          left.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlen: length [m] ≤ 1
Hm: m ∈ eval ρ base
m ∈ eval ρ base unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
Hlen: length [m] ≤ 1
Hm: m ∈ eval ρ base
m ∈ eval ρ base apply Hm.
        -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
⊆ eval ρ patt_ind_gen unfold Included.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
⊆ eval ρ patt_ind_gen intros m.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
→ m ∈ eval ρ patt_ind_gen
          rewrite -eval_patt_ind_gen_fix.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
→ m ∈ F (eval ρ patt_ind_gen)
          unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
→ m ∈ F (eval ρ patt_ind_gen) intros H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
H: m ∈ witnessed_elements_old_of_max_len (S (S len))
m ∈ F (eval ρ patt_ind_gen)
          destruct H as [l [Hwit Hlen] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
m ∈ F (eval ρ patt_ind_gen)
          unfold Included in IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
m ∈ F (eval ρ patt_ind_gen) unfold Ensembles.In in IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
m ∈ F (eval ρ patt_ind_gen)
          rewrite F_interp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          pose proof (Hwit' := Hwit).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          unfold is_witnessing_sequence in Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≤ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct Hwit as [Hlast Hl].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
l: list M
Hlast: last l = Some m
Hl: match l with
| [] => False
| m₀ :: ms =>
    m₀ ∈ eval ρ base
    ∧ Forall
        (λ x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (m₀ :: ms) ms)
end
Hlen: length l ≤ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
Hlast: last [] = Some m
Hl: False
Hlen: length [] ≤ S (S len)
Hwit': is_witnessing_sequence_old m []
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
Hlast: last [] = Some m
Hl: False
Hlen: length [] ≤ S (S len)
Hwit': is_witnessing_sequence_old m []
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) contradiction. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          
          simpl in Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: S (length l) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct Hl as [Hd Hl].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l) l)
Hlen: S (length l) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: last [d] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: last [d] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some d = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) inversion Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some d = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d]
H0: d = m
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d]
H0: d = m
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            left.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m ∈ eval ρ base unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m: M
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m ∈ eval ρ base apply Hd.
          +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m
∈ eval ρ base
  ∪ app_ext (eval ρ step) (eval ρ patt_ind_gen) right.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            pose proof (P := witnessing_sequence_old_extend).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≤ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            destruct l as [|x' l'].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
Hlast: last [d; d0] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
Hlast: last [d; d0] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
Hlast: Some d0 = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) inversion Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0: M
Hlast: Some d0 = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≤ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
H0: d0 = m
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              specialize (P d d m []).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: is_witnessing_sequence_old d [d]
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m ∈ app_interp step' d)
↔ last [d] = Some d
  ∧ is_witnessing_sequence_old m ([d] ++ [m])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              destruct P as [_ P].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: last [d] = Some d
∧ is_witnessing_sequence_old m ([d] ++ [m])
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) simpl in P.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              simpl in Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              inversion Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
x: (M * M)%type
l: list (M * M)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x = (d, m)
H0: l = []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              assert (Hsomed: Some d = Some d).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Some d = Some d
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Some d = Some d reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
→ is_witnessing_sequence_old d [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              specialize (P (conj Hsomed Hwit')).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
P: is_witnessing_sequence_old d [d]
∧ (∃ step' : M,
     step' ∈ eval ρ step ∧ m ∈ app_interp step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              destruct P as [Hwitd [step' Hstep'] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step ∧ m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step ∧ m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' re exists d.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step ∧ m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
∧ d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' d
              destruct Hstep' as [Hstep' Hm].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
∧ d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' d
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' d
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step apply Hstep'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' d
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_interp step' d
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen apply IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ witnessed_elements_old_of_max_len (S len) unfold witnessed_elements_old_of_max_len.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
                exists [d].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d] ∧ length [d] ≤ S len
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
length [d] ≤ S len
                2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
length [d] ≤ S len simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
1 ≤ S len lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]
                unfold is_witnessing_sequence.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d] simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
last [d] = Some d
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
last [d] = Some d reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
∧ Forall
    (λ x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) 
  (zip [d] [])
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base apply Hd. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [d] [])
                apply Forall_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
True
                exact I.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≤ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_interp step' d
              apply Hm.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            simpl in P.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            simpl in Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            simpl in Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            inversion Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
x: (M * M)%type
l: list (M * M)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
H: x = (d, d0)
H0: l =
(d0, x')
:: match l' with
   | [] => []
   | x2 :: l2 => (x', x2) :: zip l' l2
   end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            inversion H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
x: (M * M)%type
l: list (M * M)
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
H: x = (d0, x')
H0: l =
match l' with
| [] => []
| x2 :: l2 => (x', x2) :: zip l' l2
end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            pose (mp := (rev (d0 :: x' :: l')) !! 1).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= rev (d0 :: x' :: l') !! 1: option M
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            simpl in mp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            epose proof (Htot := (@list_lookup_lookup_total_lt (Domain M) _ (rev (d0 :: x' :: l')) 1 _)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            Unshelve.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Inhabited M
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l')) 2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Inhabited M constructor.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
M exact d0. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l'))
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l')) simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length ((rev l' ++ [x']) ++ [d0]) rewrite app_length.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l' ++ [x']) + length [d0] rewrite app_length.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l') + length [x'] + length [d0] simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l') + 1 + 1  lia.  }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            remember (@lookup_total nat (@Domain Σ M) (list (@Domain Σ M))
                 (@list_lookup_total (@Domain Σ M) {| inhabitant := d0 |}) (S O)
                 (@rev (@Domain Σ M) (@cons (@Domain Σ M) d0 (@cons (@Domain Σ M) x' l')))) as mprev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
⊆ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
mprev: M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            specialize (IHlen mprev).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ witnessed_elements_old_of_max_len (S len)
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            unfold witnessed_elements_old_of_max_len in IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: ∀ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            specialize (P mprev d m  (rev (tl (rev (d0::x'::l'))))).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            assert (Heq1: d::d0::x'::l' = d::(rev (tail (rev (d0::x'::l')))) ++ [m]).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
              apply f_equal.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d0 :: x' :: l' =
rev (tail (rev (d0 :: x' :: l'))) ++ [m]
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]
              assert (Hx'r: [x'] = rev [x']).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
[x'] = rev [x']
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
[x'] = rev [x'] reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]
              rewrite Hx'r.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ rev [x']) ++ [d0])) ++ [m]
              rewrite -rev_app_distr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]
              assert (Hd0r: [d0] = rev [d0]).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
[d0] = rev [d0]
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
[d0] = rev [d0] reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]
              rewrite Hd0r.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ rev [d0])) ++ [m]
              rewrite -rev_app_distr.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([d0] ++ [x'] ++ l'))) ++ [m]
              rewrite rev_tail_rev_app_last.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
last ([d0] ++ [x'] ++ l') = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
last (x' :: l') = Some m
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'
              apply Hlast.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'
              reflexivity.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            destruct P as [_ P].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last (d :: rev (tail (rev (d0 :: x' :: l')))) =
Some mprev
∧ is_witnessing_sequence_old m
    (d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
→ is_witnessing_sequence_old mprev
    (d :: rev (tail (rev (d0 :: x' :: l'))))
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            

            destruct (rev (tail (rev (d0 :: x' :: l')))) eqn:Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = []
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: (d1 :: l) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = []
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              apply length_zero_iff_nil in Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: length (rev (tail (rev (d0 :: x' :: l')))) =
0
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              rewrite rev_length in Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (tail (rev (d0 :: x' :: l'))) = 0
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              apply length_tail_zero in Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (rev (d0 :: x' :: l')) = 0
∨ length (rev (d0 :: x' :: l')) = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              rewrite rev_length in Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (d0 :: x' :: l') = 0
∨ length (d0 :: x' :: l') = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
              simpl in Heqrev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
→ is_witnessing_sequence_old mprev [d]
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: S (S (length l')) = 0 ∨ S (S (length l')) = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) lia.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: (d1 :: l) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            simpl in Heq1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: d1 :: l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) inversion Heq1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≤ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: d1 :: l ++ [m]
H0: d0 = d1
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) subst d0.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Heq1: d :: d1 :: x' :: l' = d :: d1 :: l ++ [m]
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Heq1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            simpl in P.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: l ++ [m])
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            rewrite -H2 in P.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            
            assert (Hm : (match l with [] => Some d1 | _ :: _ => last l end ) = Some mprev).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
              clear P Heqrev H1 H3 H4 mp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = l ++ [m]
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
              destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [] ++ [m]
Some d1 = Some mprev
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [] ++ [m]
Some d1 = Some mprev simpl in H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [m]
Some d1 = Some mprev inversion H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [m]
H0: x' = m
H1: l' = []
Some d1 = Some mprev subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
d1, m: M
IHlen: rev [d1; m] !!! 1
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ rev [d1; m] !!! 1 ∈ eval ρ patt_ind_gen
d: M
Hlast: last [m] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; m]
Hlen: S (length [d1; m]) ≤ S (S len)
Htot: rev [d1; m] !! 1 = Some (rev [d1; m] !!! 1)
H2: [m] = [m]
Some d1 = Some (rev [d1; m] !!! 1) reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev
              simpl in H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = d0 :: l ++ [m]
last (d0 :: l) = Some mprev
              inversion H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = d0 :: l ++ [m]
H0: x' = d0
H1: l' = l ++ [m]
last (d0 :: l) = Some mprev subst x' l'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: rev (d1 :: d0 :: l ++ [m]) !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
H2: d0 :: l ++ [m] = d0 :: l ++ [m]
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≤ S (S len)
last (d0 :: l) = Some mprev clear H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: rev (d1 :: d0 :: l ++ [m]) !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≤ S (S len)
last (d0 :: l) = Some mprev
              simpl in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: ((rev (l ++ [m]) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≤ S (S len)
last (d0 :: l) = Some mprev
              destruct (l ++ [m]) eqn:Hcontra.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≤ S (S len)
last (d0 :: l) = Some mprev
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Htot: ((rev (d2 :: l0) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
last (d0 :: l) = Some mprev
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≤ S (S len)
last (d0 :: l) = Some mprev pose proof (Hcontra' := @app_length _ l [m]).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≤ S (S len)
Hcontra': length (l ++ [m]) = length l + length [m]
last (d0 :: l) = Some mprev
                rewrite Hcontra in Hcontra'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≤ S (S len)
Hcontra': length [] = length l + length [m]
last (d0 :: l) = Some mprev
                simpl in Hcontra'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≤ S (S len)
Hcontra': 0 = length l + 1
last (d0 :: l) = Some mprev lia.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Htot: ((rev (d2 :: l0) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
last (d0 :: l) = Some mprev
              rewrite -Hcontra in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev (l ++ [m]) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
last (d0 :: l) = Some mprev
              rewrite rev_app_distr in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev [m] ++ rev l) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
last (d0 :: l) = Some mprev simpl in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev l ++ [d0]) ++ [d1]) !! 0 = Some mprev
last (d0 :: l) = Some mprev

              destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 0 = Some mprev
last [d0] = Some mprev
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev (d3 :: l) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 0 = Some mprev
last [d0] = Some mprev simpl in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: Some d0 = Some mprev
last [d0] = Some mprev apply Htot. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: ((rev (d3 :: l) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev
              simpl in Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev
              rewrite -Htot.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) =
(((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0
              rewrite hd_error_lookup.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) =
head (((rev l ++ [d3]) ++ [d0]) ++ [d1])
              rewrite hd_error_app.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = head ((rev l ++ [d3]) ++ [d0]) rewrite hd_error_app.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = head (rev l ++ [d3])
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≤ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d3 :: l) = head (rev l ++ [d3])
              apply last_rev_head.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
→ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (∃ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            specialize (P (@conj _ _ Hm Hwit')).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen) clear Hm Hwit'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ (∃ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp step' mprev)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            destruct P as [Hwitmprev [step' [Hstep' Hm] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
            exists step'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
∃ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp step' re exists mprev.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
step' ∈ eval ρ step
∧ mprev ∈ eval ρ patt_ind_gen
  ∧ m ∈ app_interp step' mprev
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
step' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
mprev ∈ eval ρ patt_ind_gen
∧ m ∈ app_interp step' mprev
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
step' ∈ eval ρ step apply Hstep'. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
mprev ∈ eval ρ patt_ind_gen
∧ m ∈ app_interp step' mprev
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
mprev ∈ eval ρ patt_ind_gen
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
m ∈ app_interp step' mprev
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
m ∈ app_interp step' mprev apply Hm. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
mprev ∈ eval ρ patt_ind_gen
            apply IHlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
            exists (d::d1::l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ length (d :: d1 :: l) ≤ S len
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
length (d :: d1 :: l) ≤ S len
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l) apply Hwitmprev. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
length (d :: d1 :: l) ≤ S len
            simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
S (S (length l)) ≤ S len simpl in Hlen.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
S (S (length l)) ≤ S len
            assert (Hlen': S (length l') <= len).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
S (length l') ≤ len
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
S (S (length l)) ≤ S len
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
S (length l') ≤ len lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
S (S (length l)) ≤ S len
            assert (length (l') = length (l)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
length l' = length l
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
H: length l' = length l
S (S (length l)) ≤ S len
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
length l' = length l apply (@list_len_slice _ _ _ _ _ H2). }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
H: length l' = length l
S (S (length l)) ≤ S len
            rewrite -H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≤ S len ]}
→ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≤ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (λ x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp step' mprev
Hlen': S (length l') ≤ len
H: length l' = length l
S (S (length l')) ≤ S len lia.
      Qed.

      Lemma witnessed_elements_old_included_in_interp:
        witnessed_elements_old ⊆ (@eval Σ M ρ patt_ind_gen).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old ⊆ eval ρ patt_ind_gen
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old ⊆ eval ρ patt_ind_gen
        intros x H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x ∈ witnessed_elements_old
x ∈ eval ρ patt_ind_gen
        unfold Ensembles.In in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x ∈ witnessed_elements_old
x ∈ eval ρ patt_ind_gen unfold witnessed_elements_old in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
H: x
∈ {[ m | ∃ l : list M,
           is_witnessing_sequence_old m l ]}
x ∈ eval ρ patt_ind_gen
        destruct H as [wit Hwit].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ eval ρ patt_ind_gen
        assert (H': x ∈ (witnessed_elements_old_of_max_len (length wit))).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
H': x
∈ witnessed_elements_old_of_max_len (length wit)
x ∈ eval ρ patt_ind_gen
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit) unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit) exists wit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
is_witnessing_sequence_old x wit
∧ length wit ≤ length wit split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
is_witnessing_sequence_old x wit
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
length wit ≤ length wit apply Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
length wit ≤ length wit lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
H': x
∈ witnessed_elements_old_of_max_len (length wit)
x ∈ eval ρ patt_ind_gen
        destruct wit as [|y l'] eqn:Hl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ eval ρ patt_ind_gen
        {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen unfold is_witnessing_sequence in Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen destruct Hwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
Hl: wit = []
H: last [] = Some x
H0: False
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen contradiction. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ eval ρ patt_ind_gen
        eapply witnessed_elements_old_of_max_len_included_in_interp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ witnessed_elements_old_of_max_len (S ?len)
        simpl in H'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (S (length l'))
x ∈ witnessed_elements_old_of_max_len (S ?len)
        apply H'.
      Qed.

      Definition witnessed_elements : propset (Domain M) :=
        PropSet (λ m, ∃ l, is_witnessing_sequence m l).

      Lemma witnessed_elements_old_eq_witnessed_elements :
        witnessed_elements_old = witnessed_elements.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old = witnessed_elements
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old = witnessed_elements
        rewrite -> set_eq_subseteq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old ⊆ witnessed_elements
⊆ witnessed_elements_old
        repeat rewrite -> elem_of_subseteq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
(∀ x : M,
   x ∈ witnessed_elements_old → x ∈ witnessed_elements)
∧ (∀ x : M,
     x ∈ witnessed_elements
     → x ∈ witnessed_elements_old)
        split;
          unfold witnessed_elements_old; unfold witnessed_elements; intros m; intros [l H].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
m ∈ {[ m | ∃ l : list M, is_witnessing_sequence m l ]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence m l
m
∈ {[ m | ∃ l : list M, is_witnessing_sequence_old m l ]}
        +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
m ∈ {[ m | ∃ l : list M, is_witnessing_sequence m l ]} exists (reverse l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence m (reverse l)
          apply is_witnessing_sequence_iff_is_witnessing_sequence_old_reverse.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence_old m (reverse (reverse l))
          rewrite reverse_involutive.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence_old m l
          exact H.
        +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence m l
m
∈ {[ m | ∃ l : list M, is_witnessing_sequence_old m l ]} exists (reverse l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence m l
is_witnessing_sequence_old m (reverse l)
          apply is_witnessing_sequence_iff_is_witnessing_sequence_old_reverse.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
m: M
l: list M
H: is_witnessing_sequence m l
is_witnessing_sequence m l
          apply H.
      Qed.
      
      Lemma patt_ind_gen_simpl:
        @eval Σ M ρ patt_ind_gen = witnessed_elements.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen = witnessed_elements
      Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen = witnessed_elements
        rewrite -witnessed_elements_old_eq_witnessed_elements.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen = witnessed_elements_old
        rewrite -> set_eq_subseteq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen ⊆ witnessed_elements_old
⊆ eval ρ patt_ind_gen
        split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen ⊆ witnessed_elements_old
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old ⊆ eval ρ patt_ind_gen
        +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
eval ρ patt_ind_gen ⊆ witnessed_elements_old apply interp_included_in_witnessed_elements_old.
        +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
witnessed_elements_old ⊆ eval ρ patt_ind_gen apply witnessed_elements_old_included_in_interp.
      Qed.
      
      Section injective.
        Hypothesis (Domain_eq_dec : EqDecision (Domain M)).
        Hypothesis (Hstep_total_function : @is_total_function _ M step witnessed_elements witnessed_elements ρ).
        Hypothesis (Hstep_injective : @total_function_is_injective _ M step witnessed_elements ρ).

        Hypothesis (Hbase_step_no_confusion
                    : (eval ρ base)
                        ∩ (app_ext (eval ρ step) witnessed_elements) = ∅).

        Lemma witnessed_elements_unique_seq :
          ∀ m l₁ l₂, is_witnessing_sequence m l₁ -> is_witnessing_sequence m l₂ -> l₁ = l₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
∀ (m : M) (l₁ l₂ : list M),
  is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂
        Proof.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
∀ (m : M) (l₁ l₂ : list M),
  is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂
          intros m l₁ l₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
is_witnessing_sequence m l₁
→ is_witnessing_sequence m l₂ → l₁ = l₂
          wlog: l₁ l₂ / (length l₂ <= length l₁).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
(∀ l₁ l₂ : list M,
   length l₂ ≤ length l₁
   → is_witnessing_sequence m l₁
     → is_witnessing_sequence m l₂ → l₁ = l₂)
→ is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
length l₂ ≤ length l₁
→ is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
(∀ l₁ l₂ : list M,
   length l₂ ≤ length l₁
   → is_witnessing_sequence m l₁
     → is_witnessing_sequence m l₂ → l₁ = l₂)
→ is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂ intros H Hw₁ Hw₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
l₁ = l₂
            destruct (decide (length l₁ <= length l₂)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
l: length l₁ ≤ length l₂
l₁ = l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
l₁ = l₂
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
l: length l₁ ≤ length l₂
l₁ = l₂ symmetry.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
l: length l₁ ≤ length l₂
l₂ = l₁ apply H; auto.
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
l₁ = l₂ apply H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
length l₂ ≤ length l₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
is_witnessing_sequence m l₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
is_witnessing_sequence m l₂ lia.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
is_witnessing_sequence m l₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
is_witnessing_sequence m l₂ auto.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
H: ∀ l₁ l₂ : list M,
  length l₂ ≤ length l₁
  → is_witnessing_sequence m l₁
    → is_witnessing_sequence m l₂ → l₁ = l₂
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
n: ¬ length l₁ ≤ length l₂
is_witnessing_sequence m l₂ auto.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
length l₂ ≤ length l₁
→ is_witnessing_sequence m l₁
  → is_witnessing_sequence m l₂ → l₁ = l₂
          intros Hlen12 Hw₁ Hw₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
l₁ = l₂

          assert (Hmwit: m ∈ witnessed_elements).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
m ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
m ∈ witnessed_elements exists l₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
is_witnessing_sequence m l₁ apply Hw₁. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
l₁ = l₂

          assert (Hlcom:  (common_length l₁ l₂ = length l₂)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
            destruct Hw₁ as [[lst₁ [Hlst₁ Hbase₁] ] [Hhd₁ Hfa₁] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
lst₁: M
Hlst₁: last l₁ = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head l₁ = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l₁ (tail l₁))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
            destruct l₁ as [|m₁ l₁].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₂: list M
Hlen12: length l₂ ≤ length []
lst₁: M
Hlst₁: last [] = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length [] l₂ = length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m, m₁: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head (m₁ :: l₁) = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) l₂ = length l₂
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₂: list M
Hlen12: length l₂ ≤ length []
lst₁: M
Hlst₁: last [] = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length [] l₂ = length l₂ simpl in Hlst₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₂: list M
Hlen12: length l₂ ≤ length []
lst₁: M
Hlst₁: None = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length [] l₂ = length l₂ inversion Hlst₁. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m, m₁: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head (m₁ :: l₁) = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) l₂ = length l₂
            simpl in Hhd₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m, m₁: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: Some m₁ = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) l₂ = length l₂ inversion Hhd₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m, m₁: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: Some m₁ = Some m
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
H0: m₁ = m
common_length (m :: l₁) l₂ = length l₂ subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hhd₁: Some m = Some m
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) l₂ = length l₂ clear Hhd₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) l₂ = length l₂

            destruct Hw₂ as [[lst₂ [Hlst₂ Hbase₂] ] [Hhd₂ Hfa₂] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last l₂ = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head l₂ = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l₂ (tail l₂))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) l₂ = length l₂
            destruct l₂ as [|m₂ l₂].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last [] = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head [] = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
m₂: M
l₂: list M
Hlen12: length (m₂ :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m₂ :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head (m₂ :: l₂) = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₂ :: l₂) (tail (m₂ :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m₂ :: l₂) = length (m₂ :: l₂)
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last [] = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head [] = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length [] simpl in Hlst₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: None = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head [] = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length [] inversion Hlst₂. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
m₂: M
l₂: list M
Hlen12: length (m₂ :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m₂ :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: head (m₂ :: l₂) = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₂ :: l₂) (tail (m₂ :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m₂ :: l₂) = length (m₂ :: l₂)
            simpl in Hhd₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
m₂: M
l₂: list M
Hlen12: length (m₂ :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m₂ :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: Some m₂ = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₂ :: l₂) (tail (m₂ :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m₂ :: l₂) = length (m₂ :: l₂) inversion Hhd₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
m₂: M
l₂: list M
Hlen12: length (m₂ :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m₂ :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hhd₂: Some m₂ = Some m
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₂ :: l₂) (tail (m₂ :: l₂)))
Hmwit: m ∈ witnessed_elements
H0: m₂ = m
common_length (m :: l₁) (m :: l₂) = length (m :: l₂) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hhd₂: Some m = Some m
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m :: l₂) = length (m :: l₂) clear Hhd₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m :: l₂) = length (m :: l₂)
            
            simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
(if decide (m = m) then S (common_length l₁ l₂) else 0) =
S (length l₂)
            destruct (decide (m=m)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
e: m = m
S (common_length l₁ l₂) = S (length l₂)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
n: m ≠ m
0 = S (length l₂)
            2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
n: m ≠ m
0 = S (length l₂) contradiction. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
e: m = m
S (common_length l₁ l₂) = S (length l₂)
            clear e.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
S (common_length l₁ l₂) = S (length l₂)
            apply f_equal.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length (m :: l₂) ≤ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂

            simpl in Hlen12.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂ rename Hlen12 into Hlen12'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12': S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
            assert (Hlen12: length l₂ <= length l₁).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12': S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
length l₂ ≤ length l₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12': S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
Hlen12: length l₂ ≤ length l₁
common_length l₁ l₂ = length l₂
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12': S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
length l₂ ≤ length l₁ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12': S (length l₂) ≤ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
Hlen12: length l₂ ≤ length l₁
common_length l₁ l₂ = length l₂
            clear Hlen12'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
Hlen12: length l₂ ≤ length l₁
common_length l₁ l₂ = length l₂

            remember (length l₁) as len₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ len₁
common_length l₁ l₂ = length l₂
            rewrite Heqlen₁ in Hlen12.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ length l₁
common_length l₁ l₂ = length l₂
            assert (Hlen₁ : length l₁ <= len₁).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ length l₁
length l₁ ≤ len₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ len₁
common_length l₁ l₂ = length l₂
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ length l₁
length l₁ ≤ len₁ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ len₁
common_length l₁ l₂ = length l₂
            clear Heqlen₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lst₂: M
Hlst₂: last (m :: l₂) = Some lst₂
Hbase₂: lst₂ ∈ eval ρ base
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ len₁
common_length l₁ l₂ = length l₂
            
            move: m l₁ l₂ Hlen12 Hlen₁ Hlst₁ Hlst₂ Hfa₁ Hfa₂ Hmwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new ∈ app_ext (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ = length l₂
            induction len₁; intros m l₁ l₂ Hlen12 Hlen₁ Hlst₁ Hlst₂ Hfa₁ Hfa₂ Hmwit.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ 0
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ 0
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂ destruct l₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
l₂: list M
Hlen12: length l₂ ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] l₂ = length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (d :: l₁)
Hlen₁: length (d :: l₁) ≤ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) l₂ = length l₂
              2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (d :: l₁)
Hlen₁: length (d :: l₁) ≤ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) l₂ = length l₂ simpl in Hlen₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length (d :: l₁)
Hlen₁: S (length l₁) ≤ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) l₂ = length l₂ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
l₂: list M
Hlen12: length l₂ ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] l₂ = length l₂
              destruct l₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
Hlen12: length [] ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last [m] = Some lst₂
Hfa₁, Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length [] [] = length []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] (d :: l₂) = length (d :: l₂)
              +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m: M
Hlen12: length [] ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last [m] = Some lst₂
Hfa₁, Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length [] [] = length [] reflexivity.
              +
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] (d :: l₂) = length (d :: l₂) assert (length (d::l₂) = 0).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
length (d :: l₂) = 0
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
H: length (d :: l₂) = 0
common_length [] (d :: l₂) = length (d :: l₂)
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
length (d :: l₂) = 0 simpl in Hlen12.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: S (length l₂) ≤ 0
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
length (d :: l₂) = 0 lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
H: length (d :: l₂) = 0
common_length [] (d :: l₂) = length (d :: l₂)
                rewrite H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
m, d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length []
Hlen₁: length [] ≤ 0
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: d :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₂) (tail (m :: d :: l₂)))
Hmwit: m ∈ witnessed_elements
H: length (d :: l₂) = 0
common_length [] (d :: l₂) = 0 reflexivity.
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₂) (tail (m :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ l₂ = length l₂ destruct l₂ as [|b l₂].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last [m] = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length l₁ [] = length []
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
Hlen12: length (b :: l₂) ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last [m] = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length l₁ [] = length [] rewrite common_length_l_nil.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last [m] = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
0 = length [] reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
Hlen12: length (b :: l₂) ≤ length l₁
Hlen₁: length l₁ ≤ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length l₁ (b :: l₂) = length (b :: l₂)

              destruct l₁ as [|a l'₁] eqn:Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
Heq: l₁ = []
Hlen12: length (b :: l₂) ≤ length []
Hlen₁: length [] ≤ S len₁
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: l₂) = length (b :: l₂)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: a :: l'₁) (tail (m :: a :: l'₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
Heq: l₁ = []
Hlen12: length (b :: l₂) ≤ length []
Hlen₁: length [] ≤ S len₁
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: l₂) = length (b :: l₂) simpl in Hlen12.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
Heq: l₁ = []
Hlen12: S (length l₂) ≤ 0
Hlen₁: length [] ≤ S len₁
Hlst₁: last [m] = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: l₂) = length (b :: l₂) lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: a :: l'₁) (tail (m :: a :: l'₁)))
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)


              simpl in Hfa₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) inversion Hfa₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m: M
l₁: list M
b: M
l₂: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
x: (M * M)%type
l: list (M * M)
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H: x = (m, a)
H0: l =
match l'₁ with
| [] => []
| x2 :: l2 => (a, x2) :: zip l'₁ l2
end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) clear Hfa₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              rename H1 into Hma.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) rename H2 into Hfa₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: l₂) (tail (m :: b :: l₂)))
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              simpl in Hfa₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match l₂ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip l₂ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) inversion Hfa₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match l₂ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip l₂ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
x: (M * M)%type
l: list (M * M)
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
H: x = (m, b)
H0: l =
match l₂ with
| [] => []
| x2 :: l2 => (b, x2) :: zip l₂ l2
end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match l₂ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip l₂ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) clear Hfa₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              rename H1 into Hmb.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) rename H2 into Hfa₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)

              assert (Hwita: a ∈ witnessed_elements).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
a ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
a ∈ witnessed_elements exists (a::l'₁).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
is_witnessing_sequence a (a :: l'₁)
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
∃ lst : M,
  last (a :: l'₁) = Some lst ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
head (a :: l'₁) = Some a
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (a :: l'₁) (tail (a :: l'₁)))
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
∃ lst : M,
  last (a :: l'₁) = Some lst ∧ lst ∈ eval ρ base exists lst₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
last (a :: l'₁) = Some lst₁ ∧ lst₁ ∈ eval ρ base split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
last (a :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
lst₁ ∈ eval ρ base
                  {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
last (a :: l'₁) = Some lst₁ simpl in Hlst₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
last (a :: l'₁) = Some lst₁ simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
last (a :: l'₁) = Some lst₁ apply Hlst₁. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
lst₁ ∈ eval ρ base
                  apply Hbase₁.
                }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
head (a :: l'₁) = Some a
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (a :: l'₁) (tail (a :: l'₁)))
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
head (a :: l'₁) = Some a
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l'₁) (tail (a :: l'₁)))
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
head (a :: l'₁) = Some a reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l'₁) (tail (a :: l'₁)))
                simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end apply Hfa₁.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)

              assert (Hwitb: b ∈ witnessed_elements).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
b ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
b ∈ witnessed_elements exists (b::l₂).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
is_witnessing_sequence b (b :: l₂)
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
∃ lst : M,
  last (b :: l₂) = Some lst ∧ lst ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: l₂) = Some b
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (b :: l₂) (tail (b :: l₂)))
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
∃ lst : M,
  last (b :: l₂) = Some lst ∧ lst ∈ eval ρ base exists lst₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: l₂) = Some lst₂ ∧ lst₂ ∈ eval ρ base split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
lst₂ ∈ eval ρ base
                  {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: l₂) = Some lst₂ simpl in Hlst₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: l₂) = Some lst₂ simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: l₂) = Some lst₂ apply Hlst₂. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
lst₂ ∈ eval ρ base
                  apply Hbase₂.
                }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: l₂) = Some b
∧ Forall
    (uncurry
       (λ new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (b :: l₂) (tail (b :: l₂)))
                split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: l₂) = Some b
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
                {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: l₂) = Some b reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
                simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end apply Hfa₂.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, a)
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              
              simpl in Hma.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (m, b)
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂) simpl in Hmb.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)

              assert (Ham: app_ext (eval ρ step) {[a]} = {[m]}).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}
                unfold is_total_function in Hstep_total_function.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}
                pose proof (Hstep_total_function Hwita).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
H: ∃ m₂ : M,
  m₂ ∈ witnessed_elements
  ∧ app_ext (eval ρ step) {[a]} = {[m₂]}
app_ext (eval ρ step) {[a]} = {[m]}
                destruct H as [a' [_ Ha'] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
app_ext (eval ρ step) {[a]} = {[m]}
                rewrite Ha' in Hma.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
Hma: m ∈ {[a']}
app_ext (eval ρ step) {[a]} = {[m]}
                inversion Hma.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
Hma: m ∈ {[a']}
H: a' = m
app_ext (eval ρ step) {[a]} = {[m]} subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Hma: m ∈ {[m]}
Ha': app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[a]} = {[m]}
                apply Ha'.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)

              assert (Hbm: app_ext (eval ρ step) {[b]} = {[m]}).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}
                unfold is_total_function in Hstep_total_function.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}
                pose proof (Hstep_total_function Hwitb).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
H: ∃ m₂ : M,
  m₂ ∈ witnessed_elements
  ∧ app_ext (eval ρ step) {[b]} = {[m₂]}
app_ext (eval ρ step) {[b]} = {[m]}
                destruct H as [b' [_ Hb'] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
app_ext (eval ρ step) {[b]} = {[m]}
                rewrite Hb' in Hmb.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
Hmb: m ∈ {[b']}
app_ext (eval ρ step) {[b]} = {[m]}
                inversion Hmb.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
Hmb: m ∈ {[b']}
H: b' = m
app_ext (eval ρ step) {[b]} = {[m]} subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: ∀ m₁ : M,
  m₁ ∈ witnessed_elements
  → ∃ m₂ : M,
      m₂ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[m₂]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hmb: m ∈ {[m]}
Hb': app_ext (eval ρ step) {[b]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}
                apply Hb'.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)

              assert (Haeqb: a = b).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a = b
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
Haeqb: a = b
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a = b apply Hstep_injective.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
b ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b
                apply Hwita.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
b ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b apply Hwitb.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b
                unfold rel_of.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
app_ext (eval ρ step) {[a]} =
app_ext (eval ρ step) {[b]}
                rewrite Ham.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
{[m]} = app_ext (eval ρ step) {[b]}
                rewrite Hbm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
{[m]} = {[m]}
                reflexivity.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂: list M
a: M
l'₁: list M
Hlen12: length (b :: l₂) ≤ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≤ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
Haeqb: a = b
common_length (a :: l'₁) (b :: l₂) = length (b :: l₂)
              subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hma, Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwita, Hwitb: b ∈ witnessed_elements
Ham, Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (b :: l'₁) (b :: l₂) = length (b :: l₂)
              clear Ham Hbm Hwita Hma.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
common_length (b :: l'₁) (b :: l₂) = length (b :: l₂)
              
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
(if decide (b = b)
 then S (common_length l'₁ l₂)
 else 0) = S (length l₂)
              destruct (decide (b=b)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
S (common_length l'₁ l₂) = S (length l₂)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
n: b ≠ b
0 = S (length l₂)
              2: {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
n: b ≠ b
0 = S (length l₂) contradiction. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
S (common_length l'₁ l₂) = S (length l₂)
              apply f_equal.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
common_length l'₁ l₂ = length l₂

              apply IHlen₁ with (m := b).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l₂ ≤ length l'₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≤ len₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l₂ ≤ length l'₁ simpl in Hlen12.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: S (length l₂) ≤ S (length l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l₂ ≤ length l'₁  lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≤ len₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≤ len₁ simpl in Hlen₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: S (length l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≤ len₁ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              simpl in Hlst₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements apply Hlst₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              simpl in Hlst₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l₂) = Some lst₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements apply Hlst₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements apply Hfa₁.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l₂) (tail (b :: l₂)))
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements apply Hfa₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lst₂: M
Hbase₂: lst₂ ∈ eval ρ base
len₁: nat
IHlen₁: ∀ (m : M) (l₁ l₂ : list M),
  length l₂ ≤ length l₁
  → length l₁ ≤ len₁
    → last (m :: l₁) = Some lst₁
      → last (m :: l₂) = Some lst₂
        → Forall
            (uncurry
               (λ new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          → Forall
              (uncurry
                 (λ new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: l₂) (tail (m :: l₂)))
            → m ∈ witnessed_elements
              → common_length l₁ l₂ =
                length l₂
m, b: M
l₂, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≤ S len₁
Hlen12: length (b :: l₂) ≤ length (b :: l'₁)
Hlst₂: last (m :: b :: l₂) = Some lst₂
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfa₂: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l₂ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l₂ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
              apply Hwitb.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
l₁ = l₂

          assert (Hlast12: l₁ !! (length l₂ - 1) = l₂ !! (length l₂ - 1)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
l₁ !! (length l₂ - 1) = l₂ !! (length l₂ - 1)
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
l₁ !! (length l₂ - 1) = l₂ !! (length l₂ - 1)
            eapply equal_up_to_common_length.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
length l₂ - 1 < common_length l₁ l₂
            erewrite Hlcom.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
length l₂ - 1 < length l₂
            assert (length l₂ > 0).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
length l₂ > 0
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
H: length l₂ > 0
length l₂ - 1 < length l₂
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
length l₂ > 0
              destruct l₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m (d :: l₂)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: l₂) = length (d :: l₂)
length (d :: l₂) > 0
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0 unfold is_witnessing_sequence in Hw₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: (∃ lst : M,
   last [] = Some lst ∧ lst ∈ eval ρ base)
∧ head [] = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
                simpl in Hw₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: (∃ lst : M, None = Some lst ∧ lst ∈ eval ρ base)
∧ None = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0 destruct Hw₂ as [_ [Contra _] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Contra: None = Some m
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
                inversion Contra.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m (d :: l₂)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: l₂) = length (d :: l₂)
length (d :: l₂) > 0
              simpl.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
d: M
l₂: list M
Hlen12: length (d :: l₂) ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m (d :: l₂)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: l₂) = length (d :: l₂)
S (length l₂) > 0 lia.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
H: length l₂ > 0
length l₂ - 1 < length l₂
            lia.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
l₁ = l₂

          (*assert (Hbasem': eval ρ base lst).*)          
          
          assert (~ (length l₁ > length l₂)).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
¬ length l₁ > length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
¬ length l₁ > length l₂
            intros Hcontra.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
False
            assert (Hlt : length l₂ - 1 < length l₁).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
length l₂ - 1 < length l₁
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
Hlt: length l₂ - 1 < length l₁
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
length l₂ - 1 < length l₁ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
Hlt: length l₂ - 1 < length l₁
False
            pose proof (Hexm := list_ex_elem l₁ (length l₂ - 1) Hlt).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
Hlt: length l₂ - 1 < length l₁
Hexm: ∃ x : M, l₁ !! (length l₂ - 1) = Some x
False clear Hlt.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
Hexm: ∃ x : M, l₁ !! (length l₂ - 1) = Some x
False
            destruct Hexm as [m' Hm'].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
False
            pose proof (Hwsm := witnessing_sequence_middle Hw₁ Hm').
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
False
            pose proof (Hw₂' := Hw₂).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
Hw₂': is_witnessing_sequence m l₂
False
            unfold is_witnessing_sequence in Hw₂'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
Hw₂': (∃ lst : M,
   last l₂ = Some lst ∧ lst ∈ eval ρ base)
∧ head l₂ = Some m
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip l₂ (tail l₂))
False
            destruct Hw₂' as [[lst [Hlst1 Hlst2] ] _].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst1: last l₂ = Some lst
Hlst2: lst ∈ eval ρ base
False
            rewrite list_last_length in Hlst1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
False
            assert (Hlsteqm' : m' = lst).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
m' = lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
Hlsteqm': m' = lst
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
m' = lst
              assert (Some m' = Some lst).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
Some m' = Some lst
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
H: Some m' = Some lst
m' = lst
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
Some m' = Some lst rewrite -Hm'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
l₁ !! (length l₂ - 1) = Some lst rewrite -Hlst1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
l₁ !! (length l₂ - 1) = l₂ !! (length l₂ - 1) apply Hlast12.  }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
H: Some m' = Some lst
m' = lst
              inversion H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
H: Some m' = Some lst
H1: m' = lst
lst = lst reflexivity.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: l₂ !! (length l₂ - 1) = Some lst
Hlsteqm': m' = lst
False
            subst lst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
→ is_witnessing_sequence m'
    (drop (length l₂ - 1) l₁)
Hlst1: l₂ !! (length l₂ - 1) = Some m'
Hlst2: m' ∈ eval ρ base
False
            specialize (Hwsm Hlst2).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hm': l₁ !! (length l₂ - 1) = Some m'
Hwsm: is_witnessing_sequence m'
  (drop (length l₂ - 1) l₁)
Hlst1: l₂ !! (length l₂ - 1) = Some m'
Hlst2: m' ∈ eval ρ base
False
            clear Hm' Hlst1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Hwsm: is_witnessing_sequence m'
  (drop (length l₂ - 1) l₁)
Hlst2: m' ∈ eval ρ base
False

            destruct (drop (length l₂ - 1) l₁) eqn:Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Heq: drop (length l₂ - 1) l₁ = []
Hwsm: is_witnessing_sequence m' []
Hlst2: m' ∈ eval ρ base
False
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: l
Hwsm: is_witnessing_sequence m' (d :: l)
Hlst2: m' ∈ eval ρ base
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Heq: drop (length l₂ - 1) l₁ = []
Hwsm: is_witnessing_sequence m' []
Hlst2: m' ∈ eval ρ base
False
              destruct Hwsm as [_ [Contra _] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m': M
Heq: drop (length l₂ - 1) l₁ = []
Contra: head [] = Some m'
Hlst2: m' ∈ eval ρ base
False
              inversion Contra.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: l
Hwsm: is_witnessing_sequence m' (d :: l)
Hlst2: m' ∈ eval ρ base
False
            (* if `l` is empty, then length l₁ = length l₂ -> Contradiction *)
            destruct l.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
False
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
False
              assert (Hlendrop: length (drop (length l₂ - 1) l₁) = 1).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length (drop (length l₂ - 1) l₁) = 1
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length (drop (length l₂ - 1) l₁) = 1
False
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length (drop (length l₂ - 1) l₁) = 1 rewrite Heq.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length [d] = 1 reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length (drop (length l₂ - 1) l₁) = 1
False
              rewrite drop_length in Hlendrop.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
False
              assert (length l₂ <> 0).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
length l₂ ≠ 0
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
H: length l₂ ≠ 0
False
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
length l₂ ≠ 0 intros Hcontra'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
Hcontra': length l₂ = 0
False
                apply nil_length_inv in Hcontra'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
Hcontra': l₂ = []
False
                subst l₂.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hcontra: length l₁ > length []
Hlast12: l₁ !! (length [] - 1) =
[] !! (length [] - 1)
Hlcom: common_length l₁ [] = length []
m', d: M
Heq: drop (length [] - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length [] - 1) = 1
False
                destruct Hw₂ as [_ [HContra'' _] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁: list M
Hlen12: length [] ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
HContra'': head [] = Some m
Hmwit: m ∈ witnessed_elements
Hcontra: length l₁ > length []
Hlast12: l₁ !! (length [] - 1) =
[] !! (length [] - 1)
Hlcom: common_length l₁ [] = length []
m', d: M
Heq: drop (length [] - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length [] - 1) = 1
False
                inversion HContra''.
              }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d: M
Heq: drop (length l₂ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length l₂ - 1) = 1
H: length l₂ ≠ 0
False
              lia.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
False
            (* `d0` is a member of `witnessed_elements` *)
            epose proof (Hwd0 := witnessing_sequence_tail Hwsm).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: head (tail (d :: d0 :: l)) = Some ?m'
→ is_witnessing_sequence 
    ?m' (tail (d :: d0 :: l))
False
            simpl in Hwd0.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
→ is_witnessing_sequence ?m' (d0 :: l)
False
            assert (Htmp: Some d0 = Some d0).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
→ is_witnessing_sequence ?m' (d0 :: l)
Some d0 = Some d0
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
→ is_witnessing_sequence ?m' (d0 :: l)
Htmp: Some d0 = Some d0
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
→ is_witnessing_sequence ?m' (d0 :: l)
Some d0 = Some d0 reflexivity. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
→ is_witnessing_sequence ?m' (d0 :: l)
Htmp: Some d0 = Some d0
False
            specialize (Hwd0 Htmp).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Htmp: Some d0 = Some d0
False clear Htmp.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
False

            assert (Hd0we : d0 ∈ witnessed_elements).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
d0 ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
d0 ∈ witnessed_elements exists (d0::l).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
is_witnessing_sequence d0 (d0 :: l) apply Hwd0. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False

            assert (Heqm'd : m' = d).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
Heqm'd: m' = d
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d unfold is_witnessing_sequence in Hwsm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: (∃ lst : M,
   last (d :: d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (d :: d0 :: l) = Some m'
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (d :: d0 :: l) (tail (d :: d0 :: l)))
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
              destruct Hwsm as [_ [H _] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
H: head (d :: d0 :: l) = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
              simpl in H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
H: Some d = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d inversion H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
H: Some d = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: d = m'
m' = m'
              reflexivity.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
m', d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
Heqm'd: m' = d
False
            subst m'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False

            (* `m'=d` matches `app_ext (eval ρ step) d0` *)
            assert (d ∈ app_ext
                      (eval ρ step)
                      {[d0]}).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
              unfold is_witnessing_sequence in Hwsm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: (∃ lst : M,
   last (d :: d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (d :: d0 :: l) = Some d
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (d :: d0 :: l) (tail (d :: d0 :: l)))
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
              simpl in Hwsm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: (∃ lst : M,
   last (d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ Some d = Some d
  ∧ Forall
      (uncurry
         (λ new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      ((d, d0)
       :: match l with
          | [] => []
          | x2 :: l2 => (d0, x2) :: zip l l2
          end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
              destruct Hwsm as [_ [_ Hwsm] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
              inversion Hwsm.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (d, d0)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
H: x = (d, d0)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d0, x2) :: zip l l2
end
d ∈ app_ext (eval ρ step) {[d0]} subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: uncurry
  (λ new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) 
  (d, d0)
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[d0]}
              simpl in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: d ∈ app_ext (eval ρ step) {[d0]}
H2: Forall
  (uncurry
     (λ new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[d0]}
              apply H1.
            }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False
            rewrite -> set_eq_subseteq in Hbase_step_no_confusion.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False
            destruct Hbase_step_no_confusion as [H1 _].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: eval ρ base
∩ app_ext (eval ρ step) witnessed_elements ⊆ ∅
False
            rewrite -> elem_of_subseteq in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: ∀ x : M,
  x
  ∈ eval ρ base
    ∩ app_ext (eval ρ step) witnessed_elements
  → x ∈ ∅
False
            specialize (H1 d).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
→ d ∈ ∅
False
            apply not_elem_of_empty in H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: False
False
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
→ d ∈ ∅
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
            {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: False
False exact H1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
→ d ∈ ∅
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
            clear H1.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
            split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ eval ρ base
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ eval ρ base apply Hlst2.
            -
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements unfold Ensembles.In.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements
              unfold app_ext.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d
∈ {[ e | ∃ le re : M,
           le ∈ eval ρ step
           ∧ re ∈ witnessed_elements
             ∧ e ∈ app_interp le re ]}
              destruct H as [le' [re' [H1 [H2 H3] ] ] ].
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d
∈ {[ e | ∃ le re : M,
           le ∈ eval ρ step
           ∧ re ∈ witnessed_elements
             ∧ e ∈ app_interp le re ]}
              exists le'.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
∃ re : M,
  le' ∈ eval ρ step
  ∧ re ∈ witnessed_elements ∧ d ∈ app_interp le' re exists d0.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
le' ∈ eval ρ step
∧ d0 ∈ witnessed_elements ∧ d ∈ app_interp le' d0
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
le' ∈ eval ρ step
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d0 ∈ witnessed_elements ∧ d ∈ app_interp le' d0
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
le' ∈ eval ρ step apply H1. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d0 ∈ witnessed_elements ∧ d ∈ app_interp le' d0
              split.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d0 ∈ witnessed_elements
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d ∈ app_interp le' d0
              {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d0 ∈ witnessed_elements apply Hd0we. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
d ∈ app_interp le' d0
              inversion H2.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d, d0: M
l: list M
Heq: drop (length l₂ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp le' re'
H: d0 = re'
d ∈ app_interp le' re' subst.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements ⊆ ∅
⊆ eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hcontra: length l₁ > length l₂
d: M
l: list M
re': M
Heq: drop (length l₂ - 1) l₁ = d :: re' :: l
Hlst2: d ∈ eval ρ base
Hd0we: re' ∈ witnessed_elements
Hwd0: is_witnessing_sequence re' (re' :: l)
Hwsm: is_witnessing_sequence d (d :: re' :: l)
le': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[re']}
H3: d ∈ app_interp le' re'
d ∈ app_interp le' re' apply H3.
          }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
l₁ = l₂
          assert (Hlength12: length l₁ = length l₂).
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
length l₁ = length l₂
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
Hlength12: length l₁ = length l₂
l₁ = l₂
          {
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
length l₁ = length l₂ lia. }
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hlen12: length l₂ ≤ length l₁
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
H: ¬ length l₁ > length l₂
Hlength12: length l₁ = length l₂
l₁ = l₂
          clear Hlen12 H.
Σ: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ (svar_open 0 X patt_ind_gen_body) X: propset M → propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = ∅
m: M
l₁, l₂: list M
Hw₁: is_witnessing_sequence m l₁
Hw₂: is_witnessing_sequence m l₂
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ l₂ = length l₂
Hlast12: l₁ !! (length l₂ - 1) =
l₂ !! (length l₂ - 1)
Hlength12: length l₁ = length l₂
l₁ = l₂

          apply (common_length_impl_eq _ _ Hlength12 Hlcom).
        Qed.
        
      End injective.
      
    End with_eval.

  End inductive_generation.

End with_signature.